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Differential geometry of surfaces

Differential geometry of surfaces is a branch of mathematics that investigates the properties of smooth two-dimensional manifolds embedded in three-dimensional Euclidean space, utilizing tools from multivariable calculus and linear algebra to analyze both local and global geometric features. It focuses on intrinsic aspects, such as distances, angles, and geodesic paths that can be determined solely from the surface's metric structure, as well as extrinsic aspects, like the bending and embedding of the surface in ambient space. Central to this study is the parametrization of surfaces, which allows for local representations using coordinates (u, v), enabling the computation of tangent planes and normal vectors at each point. Key concepts include the , which defines the Riemannian metric on the surface through coefficients E, F, and G to measure lengths and angles in the tangent plane, and the second fundamental form, with coefficients L, M, and N, which quantifies how the surface curves away from the tangent plane. From these, principal curvatures— the maximum and minimum curvatures at a point—are derived as eigenvalues of the shape operator, leading to Gaussian curvature (K = κ₁κ₂), an intrinsic invariant that remains unchanged under isometries, as established by Gauss's , and mean curvature (H = (κ₁ + κ₂)/2), which describes average bending and depends on the choice of normal orientation. Geodesics, the shortest paths on the surface analogous to straight lines, are characterized by parallel transport of their tangent vectors, with their behavior governed by the derived from the metric. Historically, foundational contributions include Euler's 1760 work on principal curvatures, Gauss's 1827 Disquisitiones generales circa superficies curvas, which introduced the fundamental forms and , and Bonnet's 1848 theorems on global properties, with modern developments building on these through and applications in fields like and . The subject distinguishes surfaces by properties such as minimal surfaces (H = 0), where vanishes, or surfaces of constant , like spheres (K > 0), planes (K = 0), and pseudospheres (K < 0), revealing deep connections between local differential invariants and global topology via theorems like Gauss-Bonnet.

Introduction and History

Overview

Differential geometry of surfaces is a branch of mathematics that examines the properties of smooth two-dimensional manifolds, typically embedded in three-dimensional Euclidean space \mathbb{R}^3 or studied abstractly as Riemannian manifolds equipped with a metric tensor. This field employs tools from multivariable calculus, linear algebra, and topology to analyze local and global geometric features of surfaces, such as distances, angles, and curvatures, without relying solely on their visualization in higher dimensions. A key distinction in the subject lies between intrinsic and extrinsic geometry: intrinsic properties, like Gaussian curvature, are measurable using only the surface's own metric structure and remain invariant under isometric deformations, allowing one to understand the surface's geometry independently of its embedding in \mathbb{R}^3. In contrast, extrinsic properties, such as mean curvature, depend on how the surface is positioned in the ambient space. The first and second fundamental forms provide the foundational quadratic forms that capture these aspects, respectively. Pioneering work by Gaspard Monge in the late 18th century introduced methods for describing surfaces via partial differential equations, while Carl Friedrich Gauss's 1827 treatise established the intrinsic nature of Gaussian curvature through his Theorema Egregium. The Gauss-Bonnet theorem exemplifies a profound connection between local curvature and global topology, stating that the integral of Gaussian curvature over a compact surface equals $2\pi times its Euler characteristic. Beyond pure mathematics, differential geometry of surfaces finds applications in computer graphics for modeling and rendering curved objects, in general relativity for describing the geometry of spacetime as a pseudo-Riemannian manifold, and in architecture for the design and fabrication of freeform structures with controlled curvature.

Historical Development

The study of differential geometry of surfaces began in the 18th century with foundational contributions from , who in the 1760s investigated surfaces of revolution, exploring their curvature properties and laying early groundwork for understanding surface geometry through calculus. Euler's work, as explored in his 1760 paper on the curvature of surfaces, addressed the intrinsic characteristics of such surfaces, influencing subsequent developments in the field. Building on this, Gaspard Monge advanced the subject in the late 1700s by introducing key concepts in his Application de l'analyse à la géométrie, presented to the French Academy in 1795, where he developed methods for applying analysis to geometric constructions, building upon his earlier definition of lines of curvature on surfaces in three-dimensional space. Monge's innovations, including the use of orthogonal projections, provided essential tools for visualizing and parameterizing surfaces, earning him recognition as a pioneer in differential geometry. A major milestone came in 1827 with Carl Friedrich Gauss's Disquisitiones generales circa superficies curvas, which introduced the Theorema Egregium, demonstrating that Gaussian curvature is an intrinsic property invariant under isometries, thus founding the intrinsic geometry of surfaces independent of their embedding in Euclidean space. This work shifted focus from extrinsic to intrinsic metrics, profoundly shaping the field's theoretical framework. In the 19th century, Pierre Ossian Bonnet extended these ideas with his 1867 theorem, which asserts that a surface is uniquely determined up to isometry by its first and second fundamental forms, resolving key questions on surface rigidity and reconstruction. Concurrently, Gaston Darboux contributed to global surface theory through his multi-volume Théorie des surfaces (1887–1896), developing methods for analyzing deformations, orthogonal trajectories, and integral invariants on surfaces, which broadened the scope to topological and global properties. The early 20th century saw further extensions via the uniformization theorem, independently proved by in 1907 through his work on automorphic functions and Fuchsian groups, and by using the Riemann mapping theorem generalized to Riemann surfaces, classifying all simply connected Riemann surfaces up to conformal equivalence. This theorem connected surface geometry to complex analysis, with Riemann surfaces serving as abstract models for multi-sheeted coverings and influencing higher-dimensional generalizations. Modern influences emerged prominently in 1915 when Albert Einstein incorporated differential geometry of surfaces into general relativity, using curved spacetime metrics—analogous to surface curvatures—to describe gravitational fields via the Einstein field equations. In the late 20th century, computational geometry drew on these foundations for surface modeling, with developments like Bézier surfaces (1960s) and NURBS (1970s) enabling precise approximations of curved surfaces in computer-aided design, integrating curvature analysis for applications in engineering and graphics.

Foundations of Surface Geometry

Definition of Regular Surfaces

In differential geometry, a regular surface is a subset S \subseteq \mathbb{R}^3 that locally resembles a plane in a precise manner, allowing the application of calculus tools from \mathbb{R}^2. Specifically, S is a regular surface if for every point p \in S, there exists a neighborhood V \subseteq \mathbb{R}^3 of p and a map \mathbf{r}: U \to V \cap S, where U \subseteq \mathbb{R}^2 is an open set, such that \mathbf{r} is smooth, bijective onto its image, and the differential d\mathbf{r}_q is injective for all q \in U. This injectivity ensures that the surface does not fold or degenerate locally, providing a two-dimensional structure embedded in three-dimensional space. The map \mathbf{r}(u,v) = (x(u,v), y(u,v), z(u,v)) is called a parametrization or chart of the surface, with partial derivatives \mathbf{r}_u = \frac{\partial \mathbf{r}}{\partial u} and \mathbf{r}_v = \frac{\partial \mathbf{r}}{\partial v}. The regularity condition requires that the cross product \mathbf{r}_u \times \mathbf{r}_v \neq \mathbf{0} everywhere in U, or equivalently, \|\mathbf{r}_u \times \mathbf{r}_v\| > 0, which guarantees the injectivity of the differential and that the parametrization traces a non-degenerate patch. These local parametrizations can be chosen to cover the entire surface, and overlapping charts satisfy the change-of-parameters theorem: if two parametrizations intersect, the transition map between their domains is a diffeomorphism, ensuring compatibility. An important aspect of regular surfaces is , which allows a consistent choice of unit across the surface. For a parametrization \mathbf{r}, the at a point \mathbf{r}(u,v) is given by N = \pm \frac{\mathbf{r}_u \times \mathbf{r}_v}{\|\mathbf{r}_u \times \mathbf{r}_v\|}, and the surface is orientable if a , nowhere-zero N: S \to S^2 exists such that \langle N(p), w \rangle = 0 for all s w at p, where the at p is spanned by \mathbf{r}_u and \mathbf{r}_v. Classic examples of surfaces include , , and . The unit sphere S^2 = \{ (x,y,z) \in \mathbb{R}^3 : x^2 + y^2 + z^2 = 1 \} admits parametrizations via spherical coordinates, such as \mathbf{r}(u,v) = (\sin u \cos v, \sin u \sin v, \cos u) for u \in (0,\pi), v \in (0,2\pi), satisfying the regularity condition. A , say z = 0, is parametrized by \mathbf{r}(u,v) = (u,v,0) over \mathbb{R}^2, where \mathbf{r}_u \times \mathbf{r}_v = (0,0,1) with constant norm 1. The x^2 + y^2 = 1 uses \mathbf{r}(u,v) = (\cos u, \sin u, v) for u \in (0,2\pi), v \in \mathbb{R}, yielding \|\mathbf{r}_u \times \mathbf{r}_v\| = 1 > 0.

Tangent and Normal Vectors

In differential geometry, the local geometry of a regular surface S embedded in \mathbb{R}^3 is first approximated by the tangent plane at each point p \in S. For a parametrization \mathbf{r}(u, v) of a neighborhood of p on S, the tangent space T_p S is the two-dimensional subspace of \mathbb{R}^3 spanned by the partial derivative vectors \mathbf{r}_u(p) and \mathbf{r}_v(p), which are linearly independent due to the regularity of the parametrization. These vectors represent the directions tangent to the coordinate curves on the surface passing through p, forming a basis for T_p S. The tangent space T_p S captures the first-order behavior of the surface near p, serving as the domain for the differential of the parametrization. Specifically, for any smooth curve \gamma(t) = (u(t), v(t)) on the surface with \gamma(0) = p, the velocity vector \mathbf{\dot{\gamma}}(0) lies in T_p S and is given by the linear combination \mathbf{r}_u(p) \dot{u}(0) + \mathbf{r}_v(p) \dot{v}(0). This differential form, d\mathbf{r} = \mathbf{r}_u \, du + \mathbf{r}_v \, dv, provides the first-order Taylor expansion of \mathbf{r} along curves on S, approximating the surface locally as its tangent plane. Perpendicular to T_p S is the normal line at p, spanned by the unit normal vector \mathbf{N}(p), defined as the normalization of the cross product of the basis vectors: \mathbf{N}(p) = \frac{\mathbf{r}_u(p) \times \mathbf{r}_v(p)}{\|\mathbf{r}_u(p) \times \mathbf{r}_v(p)\|}. By construction, \mathbf{N}(p) is orthogonal to both \mathbf{r}_u(p) and \mathbf{r}_v(p), ensuring that the tangent plane T_p S is perpendicular to the normal line; this orthogonality holds regardless of the choice of parametrization, as \mathbf{r}_u \times \mathbf{r}_v is independent of orientation up to sign. The inner product on T_p S, induced by the embedding in \mathbb{R}^3 and quantified by the first fundamental form, measures angles and lengths within the tangent plane.

Charts and Atlases

In differential geometry, a coordinate chart on a surface S is a pair (U, \phi), where U is an open subset of S and \phi: U \to \mathbb{R}^2 is a homeomorphism onto an open subset of \mathbb{R}^2. This local homeomorphism allows points on the surface to be assigned coordinates in the Euclidean plane, facilitating the application of calculus locally. For surfaces embedded in \mathbb{R}^3, such charts often arise as inverses of parametrizations \sigma: V \to U \subset S, where \sigma is smooth and its differential is injective, ensuring the chart captures the local geometry without singularities. An atlas for a surface S is a collection of charts \{(U_\alpha, \phi_\alpha)\}_{\alpha \in A} such that the domains U_\alpha cover S. To define a differentiable structure, the atlas must be compatible: for any two charts with overlapping domains U_\alpha \cap U_\beta \neq \emptyset, the transition map \phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta) is a C^\infty . A maximal atlas, obtained by adding all compatible charts, equips S with a , making it a C^\infty 2-manifold—a locally modeled on \mathbb{R}^2 with consistent smooth coordinate changes. This manifold structure relates to embeddings in ambient spaces like \mathbb{R}^3: a surface is an if the defining maps have injective differentials, preserving tangent spaces locally, but may self-intersect globally. An requires the immersion to also be a onto its image with the , ensuring the surface is properly without self-intersections, as in the classical treatment of surfaces.

Intrinsic and Extrinsic Geometry

First Fundamental Form

The first fundamental form provides a mathematical description of the intrinsic geometry of a regular surface in three-dimensional space, capturing how distances and angles are measured directly on the surface without reference to its embedding. Introduced by in his seminal 1827 paper Disquisitiones generales circa superficies curvas, it arises as the of the from \mathbb{R}^3 to the surface, defining a Riemannian on the at each point. For a surface parametrized by a regular map \mathbf{r}: U \subset \mathbb{R}^2 \to \mathbb{R}^3 with parameters u and v, the I is the symmetric bilinear form given by I = E \, du^2 + 2F \, du \, dv + G \, dv^2, where the coefficients are the inner products of the partial derivatives: E = \mathbf{r}_u \cdot \mathbf{r}_u, F = \mathbf{r}_u \cdot \mathbf{r}_v, and G = \mathbf{r}_v \cdot \mathbf{r}_v. These coefficients are smooth functions of u and v, reflecting the local stretching and shearing of the parametrization. A key property of the first fundamental form is its positive definiteness, which ensures that the surface is equipped with a valid Riemannian metric and corresponds to the regularity of the parametrization. Specifically, at each point, E > 0, G > 0, and the discriminant EG - F^2 > 0, implying that I(\mathbf{w}, \mathbf{w}) > 0 for all nonzero tangent vectors \mathbf{w} in the tangent plane. This condition guarantees that the partial derivatives \mathbf{r}_u and \mathbf{r}_v form a basis for the tangent space and that the metric induces a positive definite inner product, preventing degenerate or singular points on the surface. The positive definiteness is intrinsic to the surface's geometry and holds independently of the choice of parametrization, as long as it is regular. The first fundamental form enables the computation of arc lengths and angles on the surface, fundamental to understanding its intrinsic structure. For a curve \gamma(t) = \mathbf{r}(u(t), v(t)) on the surface, the arc length parameter s from t = a to t = b is s = \int_a^b \sqrt{I(\gamma'(t), \gamma'(t))} \, dt = \int_a^b \sqrt{E (u')^2 + 2F u' v' + G (v')^2} \, dt, where \gamma'(t) = u'(t) \mathbf{r}_u + v'(t) \mathbf{r}_v is the tangent vector. Similarly, the angle \theta between two tangent vectors \mathbf{u} and \mathbf{v} (or curves with those tangents) at a point is determined by the cosine formula \cos \theta = \frac{I(\mathbf{u}, \mathbf{v})}{\sqrt{I(\mathbf{u}, \mathbf{u}) I(\mathbf{v}, \mathbf{v})}}. These measurements are purely metric and do not depend on the extrinsic position of the surface in \mathbb{R}^3. The is invariant under of the surface, meaning it determines distances and angles solely through the , regardless of how the surface is embedded or reparametrized. If two surfaces are related by an —a distance-preserving —then their s coincide at corresponding points, preserving all metric properties such as arc lengths and angles. This invariance underscores the form's role in classifying surfaces up to bending without tearing or stretching, a central to Gauss's development of surface theory.

Second Fundamental Form and Shape Operator

The second fundamental form provides an extrinsic measure of how a surface bends in the ambient , capturing the second-order approximation of the surface's deviation from its tangent plane. For a regular surface parametrized by a map \mathbf{r}(u,v), with unit vector \mathbf{N}, the second fundamental form is expressed as the II = e \, du^2 + 2f \, du \, dv + g \, dv^2, where the coefficients are given by e = \mathbf{r}_{uu} \cdot \mathbf{N}, f = \mathbf{r}_{uv} \cdot \mathbf{N}, and g = \mathbf{r}_{vv} \cdot \mathbf{N}. These coefficients arise from the projection of the second partial of the parametrization onto the direction, reflecting the surface's relative to the embedding space. The second fundamental form can be extended to a symmetric bilinear form on the tangent space: for tangent vectors \mathbf{v} and \mathbf{w}, II(\mathbf{v}, \mathbf{w}) measures the relative rate of change of the normal along those directions. This form is intrinsically linked to the shape operator (also known as the Weingarten map), defined as S = -d\mathbf{N}, a linear map from the tangent space to itself that describes the differential of the Gauss map \mathbf{N}. Specifically, for a tangent vector \mathbf{v}, S(\mathbf{v}) = -\nabla_{\mathbf{v}} \mathbf{N}, where \nabla denotes the directional derivative in the ambient space, projected onto the tangent plane. The shape operator is self-adjoint with respect to the first fundamental form, meaning \langle S(\mathbf{v}), \mathbf{w} \rangle = \langle \mathbf{v}, S(\mathbf{w}) \rangle, where \langle \cdot, \cdot \rangle is the inner product induced by the first fundamental form. The relation between the second fundamental form and the shape operator is given by II(\mathbf{v}, \mathbf{w}) = \langle S(\mathbf{v}), \mathbf{w} \rangle for all tangent vectors \mathbf{v}, \mathbf{w}, establishing the second fundamental form as the first fundamental form composed with the shape operator. The eigenvalues of the shape operator, denoted \kappa_1 and \kappa_2, are the principal curvatures, representing the maximum and minimum normal curvatures of the surface at the point. These eigenvalues quantify the surface's bending in principal directions, where the second fundamental form diagonalizes. This framework, originating from Gauss's foundational work, enables the analysis of local surface geometry through linear algebra on the .

Gaussian and Mean Curvature

In , the principal curvatures \kappa_1 and \kappa_2 of a surface at a point are the eigenvalues of the shape , which quantifies how the surface bends away from its tangent plane in principal directions. The K is defined as the product of the principal curvatures, K = \kappa_1 \kappa_2. In terms of the coefficients of the I = E \, du^2 + 2F \, du \, dv + G \, dv^2 and the second fundamental form II = e \, du^2 + 2f \, du \, dv + g \, dv^2, it is given by K = \frac{eg - f^2}{EG - F^2}. $$ This invariant measures the intrinsic deviation of the surface from being flat, independent of embedding in [Euclidean space](/page/Euclidean_space). The [mean curvature](/page/Mean_curvature) $H$ is the average of [the principal](/page/The_Principal) curvatures, $H = \frac{\kappa_1 + \kappa_2}{2}$. Its expression in terms of the [fundamental](/page/Fundamental) forms is H = \frac{eG - 2fF + gE}{2(EG - F^2)}. For a unit tangent vector $v$ at a point on [the surface](/page/The_Surface), the normal [curvature](/page/Curvature) $\kappa_n$ in the [direction](/page/Direction) of $v$ is the ratio of the second [fundamental](/page/Fundamental) form to the first, $\kappa_n = \frac{II(v,v)}{I(v,v)}$. The principal curvatures are the [maximum and minimum](/page/Maximum_and_minimum) values of $\kappa_n$ over all directions. Geometrically, the sign of [Gaussian curvature](/page/Gaussian_curvature) classifies local surface geometry: $K > 0$ indicates an elliptic point, where the surface curves in the same manner in all directions, as on a [sphere](/page/Sphere) of [radius](/page/Radius) $r$ with $K = 1/r^2$; $K = 0$ denotes a parabolic point, like a [cylinder](/page/Cylinder), which is intrinsically flat; and $K < 0$ signifies a hyperbolic point, such as a saddle surface (hyperbolic paraboloid), where curvatures have opposite signs. These interpretations arise from the quadratic approximation of the surface near the point, akin to conic sections. ## Local Metric Properties ### Christoffel Symbols In the differential geometry of surfaces, the Christoffel symbols $\Gamma^k_{ij}$ represent the components of the Levi-Civita connection in a local coordinate system, capturing the intrinsic geometry induced by the Riemannian metric from the first fundamental form $g_{ij}$. These symbols were introduced by Elwin Bruno Christoffel in his 1869 work on quadratic differentials, providing a way to express the change in basis vectors along the surface. The Levi-Civita connection is uniquely determined by being torsion-free and compatible with the metric, ensuring that parallel transport preserves lengths and angles on the surface.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) The defining property arises from the metric compatibility condition, which states that the partial derivative of the metric satisfies \partial_i g_{jl} = \Gamma^k_{il} g_{kj} + \Gamma^k_{jl} g_{ki}, where indices $i, j, k, l$ run over the coordinate dimensions (typically 1 and 2 for surfaces), and summation over repeated indices is implied.[](https://www.math.purdue.edu/~arapura/preprints/diffgeom9.pdf) Solving this system yields the explicit formula for the symbols of the second kind: \Gamma^k_{ij} = \frac{1}{2} g^{km} \left( \partial_i g_{jm} + \partial_j g_{im} - \partial_m g_{ij} \right), where $g^{km}$ is the inverse metric tensor.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) Due to the torsion-free nature of the connection, the symbols exhibit symmetry in the lower indices: $\Gamma^k_{ij} = \Gamma^k_{ji}$.[](https://www.cis.upenn.edu/~cis6100/gma-v2-chap20.pdf) These symbols play a central role as the coordinate expression for the covariant derivative of the coordinate basis vectors: \nabla_{\partial_i} \partial_j = \Gamma^k_{ij} \partial_k, allowing the extension of differentiation to tensor fields while keeping results tangent to the surface.[24] For a 2-dimensional surface parametrized by coordinates $(u^1 = u, u^2 = v)$, with metric components $g_{11} = E$, $g_{12} = g_{21} = F$, $g_{22} = G$, and determinant $\Delta = EG - F^2$, the inverse components are $g^{11} = G/\Delta$, $g^{12} = g^{21} = -F/\Delta$, $g^{22} = E/\Delta$. Substituting into the general formula gives the six independent symbols (due to symmetry): \begin{align*} \Gamma^1_{11} &= \frac{G E_u - 2F F_u + F E_v}{2\Delta}, \ \Gamma^1_{12} &= \frac{G E_v - F G_u}{2\Delta}, \ \Gamma^1_{22} &= \frac{2 G F_v - G G_u - F G_v}{2\Delta}, \ \Gamma^2_{11} &= \frac{2 E F_u - E E_v - F E_u}{2\Delta}, \ \Gamma^2_{12} &= \frac{E G_u - F E_v}{2\Delta}, \ \Gamma^2_{22} &= \frac{E G_v - 2 F F_v + F G_u}{2\Delta}, \end{align*} where subscripts denote partial derivatives (e.g., $E_u = \partial_u E$). These expressions depend solely on the first fundamental form and its derivatives, underscoring their intrinsic character.[](https://www.math.purdue.edu/~arapura/preprints/diffgeom9.pdf)[](https://www.cis.upenn.edu/~cis6100/gma-v2-chap20.pdf) ### Gauss-Codazzi Equations The Gauss-Codazzi equations provide the fundamental compatibility conditions that relate the intrinsic geometry of a surface, as captured by its metric and the associated Christoffel symbols, to its extrinsic geometry in the embedding space $\mathbb{R}^3$, via the shape operator $S$ derived from the second fundamental form.[](https://www.gutenberg.org/files/36856/36856-pdf.pdf) These equations ensure that a given first fundamental form (metric) and second fundamental form can be realized as the geometry of an immersed surface only if they satisfy specific differential relations.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) The Codazzi-Mainardi equations express the symmetry of the covariant derivative of the shape operator on tangent vectors. For tangent vectors $X, Y$ to the surface, they state: (\nabla_X S)(Y) - (\nabla_Y S)(X) = 0, where $\nabla$ denotes the covariant derivative induced on the tangent bundle.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) This condition guarantees that the extrinsic bending of the surface, encoded in $S$, is compatible with the Levi-Civita connection defined by the metric. In local coordinates $(u,v)$ on the surface, with second fundamental form coefficients $e, f, g$ and Christoffel symbols $\Gamma^k_{ij}$, the equations take the form: \frac{\partial e}{\partial v} - \frac{\partial f}{\partial u} = e \Gamma^1_{12} + f (\Gamma^2_{12} - \Gamma^1_{11}) - g \Gamma^2_{11}, \frac{\partial f}{\partial v} - \frac{\partial g}{\partial u} = f \Gamma^1_{22} + g (\Gamma^2_{22} - \Gamma^1_{12}) - e \Gamma^2_{12}. [](https://lindnerdrwg.github.io/Differential-Geometry-of-Surfaces.pdf) The Gauss equation relates the Riemann curvature tensor $R$ of the surface to the shape operator, capturing how the intrinsic curvature arises from the extrinsic embedding. For tangent vectors $X, Y, Z$, it is given by: R(X, Y) Z = (S X \cdot Z) Y - (S Y \cdot Z) X, where the dot denotes the inner product on the tangent space.[24] For a surface, which is two-dimensional, this simplifies to an expression involving the [Gaussian curvature](/page/Gaussian_curvature) $K$, linking it to the extrinsic terms via the determinant of the shape operator, though the full tensor form highlights the structural relation. In coordinates, the Gauss equation appears as a component of the curvature tensor computation: \frac{\partial \Gamma^j_{kl}}{\partial x^i} - \frac{\partial \Gamma^j_{il}}{\partial x^k} + \Gamma^j_{i m} \Gamma^m_{kl} - \Gamma^j_{k m} \Gamma^m_{il} = \sum_{m,n=1}^2 (h_{k m} h_{l n} - h_{k n} h_{l m}) g^{j n}, where $h_{ij}$ are the coefficients of the [second fundamental form](/page/Second_fundamental_form) and $g^{ij}$ the inverse metric, with the right-hand side involving products that yield $K (g_{i l} g_{k j} - g_{i j} g_{k l})$ in two dimensions.[](https://homepages.uc.edu/~herronda/Diff_Geometry/Intro2DiffGeom.pdf) ### Theorema Egregium The Theorema Egregium, Latin for "remarkable theorem," is a foundational result in differential geometry stating that the Gaussian curvature $K$ of a surface is an intrinsic invariant, meaning it depends only on the metric properties of the surface as measured by the first fundamental form and remains unchanged under local isometries.[](https://www.gutenberg.org/files/36856/36856-pdf.pdf) This curvature can be explicitly computed using the coefficients $E$, $F$, $G$ of the first fundamental form $I = E\, du^2 + 2F\, du\, dv + G\, dv^2$ and their first and second partial derivatives, without reference to the embedding in ambient space.[](https://uregina.ca/~mareal/cs6.pdf) In particular, the formula for $K$ is given by K = \frac{eg - f^2}{EG - F^2}, where $e$, $f$, $g$ are the coefficients of the second fundamental form, but the theorem reveals that this expression simplifies to one involving solely $E$, $F$, $G$ and their derivatives, known as the Brioschi formula, which is a determinant of a matrix containing the second partial derivatives of the metric coefficients.[](https://www.math.drexel.edu/~tyu/Math538/Lecture9.pdf) A sketch of the proof relies on the Gauss-Codazzi equations, which relate the intrinsic Christoffel symbols (from the first fundamental form) to the extrinsic shape operator (from the second fundamental form). These equations show that the Gaussian curvature, defined extrinsically as the determinant of the shape operator, has its expression dominated by terms from the metric tensor, with all extrinsic contributions canceling out exactly.[](https://cims.nyu.edu/~yangd/papers/gauss.pdf) Specifically, substituting the Codazzi relations into the formula for $K$ yields an identity where the second fundamental form coefficients $e, f, g$ appear in both numerator and denominator in a way that they eliminate, leaving only intrinsic quantities like partial derivatives of $E, F, G$. This cancellation demonstrates that $K$ is preserved under any diffeomorphism that preserves the first fundamental form, such as bending the surface without stretching or tearing.[](https://uregina.ca/~mareal/cs6.pdf) The implications of the Theorema Egregium are profound for understanding surface geometry: for instance, a flat plane with $K = 0$ cannot be isometrically immersed into a sphere with constant positive $K > 0$, as the intrinsic curvature must match. More generally, it distinguishes classes of surfaces up to bending, enabling the study of rigidity and [embeddability](/page/Embedding) based solely on metric data. Historically, Gauss introduced this [insight](/page/Insight) in his 1827 Latin memoir *Disquisitiones generales circa superficies curvas*, where he emphasized its surprising nature by calling it "egregium," highlighting how [curvature](/page/Curvature), seemingly dependent on embedding, is actually a purely internal property of the surface.[](https://www.gutenberg.org/files/36856/36856-pdf.pdf) ## Examples of Surfaces ### Surfaces of Revolution A [surface of revolution](/page/Surface_of_revolution) is generated by rotating a [regular](/page/Regular) [plane curve](/page/Plane_curve), known as the profile or [meridian](/page/Meridian) curve, about an [axis](/page/Axis) lying in its plane, producing a surface with [rotational symmetry](/page/Rotational_symmetry) around that [axis](/page/Axis).[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) These surfaces provide explicit examples for computing the fundamental forms and curvatures in [differential geometry](/page/Differential_geometry), as their parametrizations simplify the relevant tensor calculations.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) The standard parametrization of a surface of revolution is given by \mathbf{r}(u,v) = \bigl( f(u) \cos v, , f(u) \sin v, , g(u) \bigr), where $ (u,v) $ are coordinates with $ u \in I $ an [interval](/page/Interval) and $ v \in [0, 2\pi) $, and $ (f(u), g(u)) $ traces the profile curve in the $ xz $-plane with $ f(u) > 0 $ to avoid self-intersections.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) The $ u $-curves are meridians (copies of the [profile](/page/Profile)), and the $ v $-curves are parallels (circles of radius $ f(u) $).[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) Assuming the profile is regular, so $ f'(u)^2 + g'(u)^2 > 0 $, the [first fundamental form](/page/First_fundamental_form) is I = \bigl( f'(u)^2 + g'(u)^2 \bigr) , du^2 + f(u)^2 , dv^2. This orthogonal metric reflects the [rotational symmetry](/page/Rotational_symmetry), with no $ du \, dv $ cross term.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) The Gaussian curvature $ K $ can be computed using the first and second fundamental forms, but for surfaces of revolution, explicit formulas arise from the principal curvatures: the meridional curvature (along the profile) and the azimuthal curvature (along parallels).[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) In general, K = -\frac{g'(u) \left( f''(u) g'(u) - f'(u) g''(u) \right)}{f(u) \bigl( f'(u)^2 + g'(u)^2 \bigr)^{2}}. This expression follows from $ K $ as the product of principal curvatures, with the meridional principal curvature $ \kappa_1 = \frac{f''(u) g'(u) - f'(u) g''(u)}{\bigl( f'(u)^2 + g'(u)^2 \bigr)^{3/2}} $ and the azimuthal $ \kappa_2 = -\frac{g'(u)}{f(u) \sqrt{f'(u)^2 + g'(u)^2}} $.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) If the profile is parametrized by arc length, so $ f'(u)^2 + g'(u)^2 = 1 $, the first fundamental form simplifies to $ I = du^2 + f(u)^2 \, dv^2 $, and the Gaussian curvature reduces to K = -\frac{f''(u)}{f(u)}. This form highlights how $ K $ depends intrinsically on the metric via the Theorema Egregium.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) Prominent examples illustrate these computations. The unit sphere is a surface of revolution with profile the semicircle $ f(u) = \sin u $, $ g(u) = \cos u $ for $ u \in (0, \pi) $, satisfying arc-length parametrization since $ f'^2 + g'^2 = 1 $.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) Here, $ f''(u) = -\sin u $, so $ K = - (-\sin u)/\sin u = 1 $, constant positive curvature as expected for a sphere.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) The catenoid, generated by rotating a catenary $ f(u) = \cosh u $, $ g(u) = u $ (with scale parameter 1), has first fundamental form $ I = \cosh^2 u \, du^2 + \cosh^2 u \, dv^2 $.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) Its Gaussian curvature is $ K = -\sech^4 u $, negative and varying, though it achieves zero mean curvature, making it a minimal surface.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) The pseudosphere, formed by rotating a tractrix $ f(u) = \sech u $, $ g(u) = u - \tanh u $, exhibits constant Gaussian curvature $ K = -1 $, serving as a model for hyperbolic geometry despite its limited extent (cusp singularity).[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) ### Quadric Surfaces Quadric surfaces in $\mathbb{R}^3$ are defined as the zero sets of [quadratic](/page/Quadratic) polynomials, given by the general [equation](/page/Equation) $Ax^2 + By^2 + Cz^2 + Dxy + Exz + Fyz + Gx + Hy + Iz + J = 0$, where $A, B, \dots, J$ are constants.[](https://www.geometrictools.com/Documentation/ClassifyingQuadrics.pdf) These surfaces arise as level sets of [quadratic](/page/Quadratic) forms and represent the simplest non-linear surfaces studied in [differential geometry](/page/Differential_geometry). To analyze their geometry, the equation is typically transformed to principal axes via an orthogonal change of coordinates, diagonalizing the associated [symmetric matrix](/page/Symmetric_matrix) of the quadratic terms, yielding forms like $ax^2 + by^2 + cz^2 = 1$ for bounded surfaces or $z = ax^2 + by^2$ for unbounded paraboloids.[](https://www.geometrictools.com/Documentation/ClassifyingQuadrics.pdf) The [classification](/page/Classification) of [quadric](/page/Quadric) surfaces relies on the eigenvalues $\lambda_1, \lambda_2, \lambda_3$ of the 3×3 [symmetric matrix](/page/Symmetric_matrix) corresponding to the quadratic terms $ \mathbf{x}^T Q \mathbf{x} $, along with the linear and constant terms after [completing the square](/page/Completing_the_square). For the non-degenerate case (all eigenvalues nonzero), the surface type is determined by the signs of the eigenvalues: all positive yields an [ellipsoid](/page/Ellipsoid); two positive and one negative yields a [hyperboloid](/page/Hyperboloid) of one sheet; one positive and two negative yields a [hyperboloid](/page/Hyperboloid) of two sheets. Degenerate cases with rank 2 (one zero eigenvalue) include elliptic paraboloids (two positive eigenvalues) and hyperbolic paraboloids (one positive, one negative); [rank 1](/page/Rank_1) cases yield cylinders or planes. This eigenvalue-based [classification](/page/Classification) captures the topological and geometric distinctions, such as [compactness](/page/Compact_space) for ellipsoids versus hyperbolicity for hyperboloids.[](https://www.geometrictools.com/Documentation/ClassifyingQuadrics.pdf) The intrinsic geometry of [quadric](/page/Quadric) surfaces is induced from the [Euclidean](/page/Euclidean) [metric](/page/Metric) on $\mathbb{R}^3$, restricting the standard [dot product](/page/Dot_product) to the tangent spaces. For example, [the sphere](/page/The_Sphere), a special quadric surface ([ellipsoid](/page/Ellipsoid) with $a = b = c = r$), parametrized in spherical coordinates as $\mathbf{r}(\theta, \phi) = (r \sin\theta \cos\phi, r \sin\theta \sin\phi, r \cos\theta)$, has [first fundamental form](/page/First_fundamental_form) $ds^2 = r^2 (d\theta^2 + \sin^2\theta \, d\phi^2)$, enabling computation of lengths and areas intrinsically.[](https://faculty.sites.iastate.edu/jia/files/inline-files/surface-curves.pdf) Similar parametrizations apply to other quadrics, such as ellipsoids via scaled spherical coordinates or hyperboloids via [hyperbolic functions](/page/Hyperbolic_functions), preserving the embedded [metric](/page/Metric) [structure](/page/Structure). Regarding extrinsic properties, the Gaussian curvature $K$ for an ellipsoid $\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1$ (with $a \geq b \geq c > 0$) is given by K(x,y,z) = \frac{1}{a^2 b^2 c^2 \left( \frac{x^2}{a^4} + \frac{y^2}{b^4} + \frac{z^2}{c^4} \right)^2}, which is positive everywhere, confirming elliptic geometry, with maximum $K = (a/(bc))^2$ at the vertices along the longest axis and minimum $K = (c/(ab))^2$ at the ends of the shortest axis.[](https://www.johndcook.com/blog/2019/10/07/curvature-of-an-ellipsoid/) For hyperboloids, $K$ changes sign, reflecting saddle-like regions, while paraboloids exhibit $K = 0$ along rulings or negative values for hyperbolic types. The total Gaussian curvature integrated over closed quadrics like the ellipsoid equals $4\pi$, consistent with the Gauss-Bonnet theorem for orientable surfaces.[](https://faculty.sites.iastate.edu/jia/files/inline-files/gaussian-curvature.pdf) ### Minimal and Ruled Surfaces Minimal surfaces are surfaces with zero [mean curvature](/page/Mean_curvature), a condition that characterizes them as critical points of the area functional, making them local minimizers of area among surfaces with fixed boundaries.[](https://www.ugr.es/~jperez/papers/bamsJan11.pdf) This variational property arises from the first variation of the area being zero, leading to the mean curvature vanishing everywhere.[](https://www.ugr.es/~jperez/papers/bamsJan11.pdf) In the context of graphs over a domain in the plane, a surface defined by $ z = r(x,y) $ is minimal if it satisfies the nonlinear [elliptic partial differential equation](/page/Elliptic_partial_differential_equation) \nabla \cdot \left( \frac{\nabla r}{\sqrt{1 + |\nabla r|^2}} \right) = 0, which ensures the mean curvature $ H = 0 $.[](https://arxiv.org/pdf/2101.01375) Prominent examples include the [catenoid](/page/Catenoid), discovered by Leonhard Euler in 1744 and verified as [minimal](/page/Minimal_surface) by Jean-Baptiste Meusnier in 1776, which is the unique complete [embedded](/page/Embedded) [minimal surface](/page/Minimal_surface) of [genus](/page/Genus) zero with two ends.[](https://www.ugr.es/~jperez/papers/bamsJan11.pdf) The [catenoid](/page/Catenoid) can be parametrized as $ x = c \cosh(v/c) \cos u $, $ y = c \cosh(v/c) \sin u $, $ z = v $, where its principal curvatures are opposites, yielding zero [mean curvature](/page/Mean_curvature).[](https://www.math.mcgill.ca/gantumur/math580f12/minimal_surfaces.pdf) The [helicoid](/page/Helicoid), also proved [minimal](/page/Minimal_surface) by Meusnier in 1776, is a simply connected [embedded](/page/Embedded) [minimal surface](/page/Minimal_surface) of [genus](/page/Genus) zero with one end, parametrized by $ x = \rho \cos \alpha \theta $, $ y = \rho \sin \alpha \theta $, $ z = \theta $, and features principal curvatures that sum to zero.[](https://www.ugr.es/~jperez/papers/bamsJan11.pdf)[](https://www.math.mcgill.ca/gantumur/math580f12/minimal_surfaces.pdf) Enneper's surface, discovered by Alfred Enneper in 1864, is another complete [minimal surface](/page/Minimal_surface) of [genus](/page/Genus) zero with one end and finite total [curvature](/page/Curvature), given parametrically by $ x = r \cos \phi - \frac{1}{3} r^3 \cos 3\phi $, $ y = -\frac{1}{3} r (r^2 \sin 3\phi + 3 \sin \phi) $, $ z = r^2 \cos^2 \phi $, though it is self-intersecting.[](https://www.ugr.es/~jperez/papers/bamsJan11.pdf)[](https://www.math.mcgill.ca/gantumur/math580f12/minimal_surfaces.pdf) Ruled surfaces are surfaces that can be expressed as the union of straight lines, known as rulings, parametrized generally as $ \mathbf{x}(u,v) = \mathbf{a}(u) + v \mathbf{r}(u) $, where $ \mathbf{r}(u) $ directs the lines.[](https://www.geometrie.tugraz.at/wallner/kurs.pdf) A special class, developable surfaces, are ruled surfaces with zero [Gaussian curvature](/page/Gaussian_curvature) $ K = 0 $, allowing them to be isometrically mapped onto the [plane](/page/Plane) without distortion; these maintain a constant [tangent](/page/Tangent) [plane](/page/Plane) along each ruling.[](https://www.geometrie.tugraz.at/wallner/kurs.pdf) Examples of developable surfaces include cones, where rulings intersect at a [vertex](/page/Vertex); cylinders, with parallel rulings; and tangent developables, generated by the [tangent](/page/Tangent) lines to a space curve $ \mathbf{a}(u) $ via $ \mathbf{x}(u,v) = \mathbf{a}(u) + v \dot{\mathbf{a}}(u) $.[](https://www.geometrie.tugraz.at/wallner/kurs.pdf) The [helicoid](/page/Helicoid) and [catenoid](/page/Catenoid) are linked as conjugate minimal [surfaces](/page/Minimal_surface), sharing the same Gauss map but differing by a 90-degree [rotation](/page/Rotation) in their height differentials, a relation established through the Bonnet transformation that preserves minimality.[](https://www.ugr.es/~jperez/papers/bamsJan11.pdf) Notably, the [helicoid](/page/Helicoid) is the only non-planar ruled minimal surface, consisting entirely of straight-line rulings while maintaining zero [mean curvature](/page/Mean_curvature).[](https://www.ugr.es/~jperez/papers/bamsJan11.pdf) ## Geodesic Geometry ### Definition and Properties of Geodesics In the differential geometry of surfaces, [a geodesic](/page/Geodesic) is [a smooth](/page/Smooth) [curve](/page/Curve) on [a Riemannian surface](/page/Riemannian_surface) that locally minimizes the arc length between nearby points, serving as the intrinsic analogue of [a straight line](/page/Straight_line) in the Euclidean plane. Formally, for a parametrized curve $\gamma: I \to S$ on [a surface](/page/Surface) $S$ with the first fundamental form defining the metric, $\gamma$ is [a geodesic](/page/Geodesic) if it is parametrized by arc length $s$ (so $\|\gamma'(s)\| = 1$) and its tangent vector field $\gamma'$ is parallel along $\gamma$, meaning the covariant derivative satisfies $\nabla_{\gamma'} \gamma' = 0$. This condition ensures that the acceleration of the curve lies in the normal direction to the surface, preventing any tangential deviation that would increase the length. The [geodesic](/page/Geodesic) equation provides the local coordinate expression for this definition. In a coordinate [chart](/page/Chart) $(u^1, u^2)$ on $S$, if $\gamma(s) = (u^1(s), u^2(s))$, the equation becomes \frac{d^2 u^k}{ds^2} + \Gamma^k_{ij} \frac{du^i}{ds} \frac{du^j}{ds} = 0, \quad k = 1,2, where repeated indices imply summation, and $\Gamma^k_{ij}$ are the [Christoffel symbols](/page/Christoffel_symbols) of the second kind, determined by the [metric tensor](/page/Metric_tensor) $g_{ij}$ via $\Gamma^k_{ij} = \frac{1}{2} g^{kl} (\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij})$. These symbols encode the [Levi-Civita connection](/page/Levi-Civita_connection) compatible with the metric, ensuring the equation is intrinsic to [the surface](/page/The_Surface). Solutions to this second-order system of [ordinary](/page/Ordinary) [differential](/page/Differential) equations describe all geodesics on $S$. Key properties follow from the existence and uniqueness theorems for solutions to the geodesic equation. For any point $p \in S$ and nonzero tangent vector $w \in T_p S$, there exists a unique geodesic $\gamma: (-\varepsilon, \varepsilon) \to S$ (for some $\varepsilon > 0$) such that $\gamma(0) = p$ and $\gamma'(0) = w / \|w\|$, assuming arc-length parametrization. This geodesic can be maximally extended until it escapes any compact set or reaches a boundary. Additionally, since the covariant derivative preserves the metric, the arc-length parametrization implies constant speed: $\|\gamma'(s)\| = 1$ for all $s$, reflecting a conservation law akin to energy preservation along the curve. Parallel transport along $\gamma$ further ensures that inner products of tangent vectors remain constant, maintaining angles and lengths intrinsically. Representative examples illustrate these concepts. On the [Euclidean plane](/page/Euclidean_plane), equipped with the flat metric $ds^2 = du^2 + dv^2$, the [Christoffel symbols](/page/Christoffel_symbols) vanish ($\Gamma^k_{ij} = 0$), so the geodesic equation reduces to $\frac{d^2 u^k}{ds^2} = 0$, yielding straight lines as the unique length-minimizing curves. On the unit sphere $S^2 \subset \mathbb{R}^3$ with the induced metric, great circles—intersections of the sphere with planes through the origin—are geodesics, as their parametrizations satisfy the equation and minimize distances globally up to antipodal points. ### Geodesic Curvature In differential geometry of surfaces, [geodesic curvature](/page/Geodesic) quantifies the deviation of a [curve](/page/Curve) from being a [geodesic](/page/Geodesic) within the surface's tangent plane, providing an extrinsic measure distinct from the intrinsic [Gaussian curvature](/page/Gaussian_curvature) of the surface. For a regular [curve](/page/Curve) $\gamma$ immersed in a surface $S \subset \mathbb{R}^3$ with unit normal $N$, let $T = \gamma' / \|\gamma'\| $ denote the unit tangent vector to $\gamma$. The [geodesic curvature](/page/Geodesic) $\kappa_g$ at a point is defined as $\kappa_g = \|\nabla_{\gamma'} T \times N\|$, where $\nabla$ is the Levi-Civita [covariant derivative](/page/Covariant_derivative) induced on the surface; this expression captures the magnitude of the tangential component of the [acceleration](/page/Acceleration) vector projected appropriately via the cross product with the normal.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf)[](https://www.math.cuhk.edu.hk/course_builder/2223/math4030/12Geodesic-1-beam-2022.pdf) A fundamental relation connects [geodesic](/page/Geodesic) curvature to the [geometry](/page/Geometry) of the [curve](/page/Curve) as [embedded](/page/Embedded) in $\mathbb{[R](/page/R)}^3$: for a unit-speed [curve](/page/Curve), $\kappa_g^2 + \kappa_n^2 = \kappa^2$, where $\kappa = \|\gamma''\|$ is the ordinary [curvature](/page/Curvature) of $\gamma$ in space, and $\kappa_n$ is the normal [curvature](/page/Curvature) measuring bending orthogonal to the tangent plane.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) The normal [curvature](/page/Curvature) $\kappa_n$ arises from the [shape operator](/page/Operator) $S$ of the surface, defined by $S(X) = -\nabla_X N$ for tangent vectors $X$, via $\kappa_n = \langle S(T), T \rangle$; this ties [geodesic](/page/Geodesic) [curvature](/page/Curvature) to the surface's [second fundamental form](/page/Second_fundamental_form) without altering its intrinsic nature.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) Geodesic curvature vanishes identically for [geodesics](/page/Geodesic), curves that locally minimize length and thus accelerate solely in the [normal](/page/Normal) direction to the surface.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) It changes sign under reversal of the [curve](/page/Curve)'s orientation or the surface's [normal](/page/Normal) but retains the same [absolute value](/page/Absolute_value) for surfaces [tangent](/page/Tangent) along the [curve](/page/Curve). For curves on surfaces, the Frenet-Serret [framework](/page/Framework) generalizes to the [Darboux frame](/page/Darboux_frame) $\{T, g, N\}$, where $g = N \times T$ is the unit geodesic principal [normal](/page/Normal) in the [tangent](/page/Tangent) [plane](/page/Plane); the structure equations are \frac{dT}{ds} = \kappa_g g, \quad \frac{dg}{ds} = -\kappa_g T + \tau_g N, \quad \frac{dN}{ds} = -\tau_g g - \kappa_n T, with geodesic torsion $\tau_g$ and $\kappa_n$ involving the shape operator as above.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf)[](https://www.math.cuhk.edu.hk/course_builder/2223/math4030/12Geodesic-1-beam-2022.pdf) This decomposition highlights how geodesic [curvature](/page/Curvature) governs turning confined to the [tangent](/page/Tangent) [plane](/page/Plane), independent of the ambient embedding. ### Orthogonal and Geodesic Polar Coordinates In differential geometry of surfaces, orthogonal coordinates refer to a parametrization $x(u,v)$ of a surface $S$ where the coordinate curves (lines of constant $u$ or $v$) intersect at right angles, meaning the [first fundamental form](/page/First_fundamental_form) satisfies $F = 0$.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) This condition simplifies the [Christoffel symbols](/page/Christoffel_symbols), as the off-diagonal metric components vanish, reducing the geodesic equations to $\Gamma^u_{uu} = \frac{1}{2E} \frac{\partial E}{\partial u}$, $\Gamma^v_{vv} = -\frac{1}{2G} \frac{\partial G}{\partial v}$, and similar terms without mixed derivatives.[](https://www.math.stonybrook.edu/~anderson/mat362-spr15/petersen.pdf) Such coordinates exist locally around any point on $S$ by choosing an orthonormal frame in the [tangent space](/page/Tangent_space) and integrating along the integral curves.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) Geodesic polar coordinates extend this orthogonality by centering the system at a point $p \in S$, with the radial coordinate $u$ measuring geodesic distance from $p$ and the angular coordinate $v$ parametrizing directions in the tangent space $T_p S$.[](https://people.math.harvard.edu/~knill/teaching/math136/handouts/lecture12.pdf) These coordinates are constructed using the exponential map $\exp_p: T_p S \to S$, which sends a tangent vector $w \in T_p S$ to the point $\exp_p(w) = \gamma_w(1)$ on the unique geodesic $\gamma_w$ starting at $p$ with initial velocity $w$, provided $|w|$ is small enough to stay within a normal neighborhood.[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) In these coordinates, the metric takes the form ds^2 = du^2 + G(u,v)^2 , dv^2, where $G(0,v) = 0$ and $\frac{\partial G}{\partial u}(0,v) = 1$, ensuring the radial lines ($v =$ constant) are unit-speed geodesics orthogonal to the coordinate circles ($u =$ constant).[](https://people.math.harvard.edu/~knill/teaching/math136/handouts/lecture12.pdf) The exponential map is a local diffeomorphism near the origin in $T_p S$, allowing this coordinate system to cover a disk-like neighborhood around $p$.[](https://www.math.stonybrook.edu/~anderson/mat362-spr15/petersen.pdf) Gauss's lemma underpins the orthogonality in geodesic polar coordinates by asserting that, within the normal neighborhood, the radial geodesics from $p$ remain perpendicular to the level sets of the distance function (the geodesic circles).[](http://home.ustc.edu.cn/~huangty0407/DGNotes/Do%20Carmo%20-%20Differential%20Geometry%20of%20Curves%20and%20Surfaces.pdf) Specifically, for a geodesic $\gamma(s) = \exp_p(s \hat{v})$ with unit initial velocity $\hat{v}$, the inner product $\langle \gamma'(s), X(\gamma(s)) \rangle = 0$ holds for any vector field $X$ tangent to the geodesic circle at distance $s$, preserving the metric's diagonal form.[](https://people.math.harvard.edu/~knill/teaching/math136/handouts/lecture12.pdf) This property follows from the conservation of the radial component of the velocity along geodesics and ensures that the coordinate system simplifies computations of geodesic curvature and lengths near $p$.[](https://www.math.stonybrook.edu/~anderson/mat362-spr15/petersen.pdf) ## Curvature Theorems ### Gauss-Bonnet Theorem The Gauss-Bonnet theorem is a cornerstone result in differential geometry that establishes a profound connection between the intrinsic geometry of a surface, quantified by its Gaussian curvature, and the surface's topological invariants.[](https://www.gutenberg.org/files/36856/36856-pdf.pdf) First articulated by Carl Friedrich Gauss for geodesic triangles in his seminal 1827 work on curved surfaces, and generalized by Pierre Ossian Bonnet in 1848 to arbitrary regions with piecewise smooth boundaries, the theorem reveals how local curvature properties integrate to determine global topological features, such as the Euler characteristic.[](https://www.gutenberg.org/files/36856/36856-pdf.pdf)[](https://mathshistory.st-andrews.ac.uk/Biographies/Bonnet/) This linkage underscores the theorem's role in bridging differential and topological aspects of surfaces, influencing developments from classical geometry to modern Riemannian geometry.[](https://www.worldscientific.com/doi/pdf/10.1142/9789812812834_0033) In its local form, the theorem applies to a compact region $D$ on an oriented Riemannian surface with piecewise smooth boundary $\partial D$. It states that the integral of the Gaussian curvature $K$ over $D$, plus the integral of the geodesic curvature $\kappa_g$ along the boundary $\partial D$, equals $2\pi$ times the Euler characteristic $\chi(D)$ of $D$: \int_D K , dA + \int_{\partial D} \kappa_g , ds = 2\pi \chi(D). This formula captures the total turning of geodesics and the accumulated curvature within the region, where $\chi(D)$ is computed via a triangulation of $D$ as vertices minus edges plus faces.[](https://www2.math.upenn.edu/~shiydong/Math501X-7-GaussBonnet.pdf) The geodesic curvature $\kappa_g$ measures the deviation of the boundary curve from a geodesic on the surface.[](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Rotskoff.pdf) For the global form, consider a compact, orientable Riemannian surface $S$ without boundary. The theorem simplifies to the total Gaussian curvature integral equaling $2\pi$ times the Euler characteristic of the entire surface: \int_S K , dA = 2\pi \chi(S). Here, $\chi(S)$ reflects the surface's topology—for instance, $\chi(S) = 2 - 2g$ for a genus-$g$ surface.[](https://arxiv.org/pdf/1701.01666) This integral form implies that surfaces with the same topology must have the same total curvature, independent of their specific embedding or metric, as long as the Gaussian curvature is defined intrinsically.[](https://www2.math.upenn.edu/~shiydong/Math501X-7-GaussBonnet.pdf) A sketch of the proof for geodesic triangles, which extends to general regions via triangulation, relies on parallel transport. Consider a small geodesic triangle on the surface with vertices $P$, $Q$, and $R$. Parallel transporting a [tangent vector](/page/Tangent_vector) around the triangle's [boundary](/page/Boundary) yields a [rotation](/page/Rotation) by the enclosed [curvature](/page/Curvature) [integral](/page/Integral), due to the [holonomy](/page/Holonomy) induced by the surface's [connection](/page/Connection). The exterior angles at the vertices, adjusted for this [holonomy](/page/Holonomy), sum to $2\pi$ minus the integrated [Gaussian curvature](/page/Gaussian_curvature) over the triangle, aligning with the theorem's local form for $\chi = 1$.[](https://arxiv.org/pdf/1701.01666) This approach highlights the theorem's intrinsic nature, provable without reference to the ambient space.[](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2010/REUPapers/Rotskoff.pdf) Applications of the theorem abound in classifying surfaces. For the unit [sphere](/page/Sphere) $S^2$, which has $\chi(S^2) = 2$, the total [curvature](/page/Curvature) is $4\pi$, consistent with its constant [Gaussian curvature](/page/Gaussian_curvature) of 1.[](https://www2.math.upenn.edu/~shiydong/Math501X-7-GaussBonnet.pdf) Similarly, for a [torus](/page/Torus) with $\chi = 0$, the total [curvature](/page/Curvature) vanishes, implying regions of positive and negative [curvature](/page/Curvature) must balance. These examples illustrate how the [theorem](/page/Theorem) constrains possible surface geometries based on [topology](/page/Topology).[](https://arxiv.org/pdf/1701.01666) ### Surfaces of Constant Curvature Surfaces with constant Gaussian curvature $K$ play a central role in the classification of Riemannian geometries on two-dimensional manifolds, as their intrinsic geometry is determined solely by the sign and magnitude of $K$. By Gauss's Theorema Egregium, $K$ is an intrinsic invariant, meaning it remains unchanged under isometries, allowing such surfaces to be locally isometric to standard model spaces: the Euclidean plane for $K = 0$, the sphere for $K > 0$, and the hyperbolic plane for $K < 0$. These models exhibit distinct behaviors in terms of geodesic divergence and parallel transport, with positive $K$ leading to focusing geodesics, zero $K$ to parallel geodesics, and negative $K$ to diverging geodesics.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) For $K = 0$, surfaces are developable, meaning they can be flattened onto the plane without distortion, as they are locally [isometric](/page/Isometric) to the Euclidean plane. Such surfaces have vanishing Gaussian curvature everywhere and are precisely the ruled surfaces where the tangent plane is constant along each ruling line. Representative examples include the plane itself, which has the flat metric $ds^2 = du^2 + dv^2$, and the cylinder, obtained by rolling a plane into a tube, preserving $K = 0$ and allowing unrolling without stretching. Cones also qualify, as their lateral surface unfolds into a sector of the plane.[](https://www.cis.upenn.edu/~cis6100/gma-v2-chap20.pdf)[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) Surfaces with constant positive Gaussian curvature $K > 0$ are locally isometric to a [sphere](/page/Sphere) of [radius](/page/Radius) $1/\sqrt{K}$. The standard example is the [sphere](/page/Sphere) of [radius](/page/Radius) $R$, where $K = 1/R^2$, embodying [spherical geometry](/page/Spherical_geometry) with closed geodesics (great circles) that reconverge after a finite [length](/page/Length). In this geometry, the elliptic plane arises as the quotient of the [sphere](/page/Sphere) by antipodal identification, yielding a simply connected model with the same local properties but different global [topology](/page/Topology). These surfaces exhibit positive total [curvature](/page/Curvature), consistent with the [Gauss-Bonnet theorem](/page/Theorem) applied to compact domains.[](https://www.cis.upenn.edu/~cis6100/gma-v2-chap20.pdf)[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) For constant negative Gaussian curvature $K < 0$, without loss of generality $K = -1$, the model is the hyperbolic plane, which admits a metric in geodesic polar coordinates given by ds^2 = du^2 + \sinh^2 u , dv^2, where $u \geq 0$ is the radial distance and $v$ is the angular coordinate, reflecting exponential growth in area elements. A concrete realization in $\mathbb{R}^3$ is the pseudosphere, formed by rotating a tractrix curve around its asymptote, yielding a surface with $K = -1$ but only covering a portion of the hyperbolic plane due to a singular cusp. The tractrix, defined as the path of an object dragged by a string of constant length along a straight line, generates this "horn" shape. However, Hilbert's theorem establishes that no complete isometric immersion of the entire hyperbolic plane exists in $\mathbb{R}^3$, as any such attempt leads to asymptotic behavior incompatible with bounded extrinsic curvature.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf)[](https://www.ms.uky.edu/~droyster/courses/fall96/math3181/notes/hyprgeom/node69.html)[](https://math.uchicago.edu/~may/REU2020/REUPapers/Dewhurst.pdf) ### Uniformization Theorem The uniformization theorem states that every Riemann surface admits a complete conformal metric of constant curvature, which classifies the surface according to its topology: the sphere (genus 0) admits a metric of positive constant curvature, the torus (genus 1) admits a flat metric of zero curvature, and surfaces of genus $ g \geq 2 $ admit a metric of negative constant curvature.[](https://eudml.org/doc/58678) This result implies that any orientable surface can be endowed with one of the three standard geometric structures—spherical, Euclidean, or hyperbolic—up to conformal equivalence.[](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/unif7.pdf) The proof of the theorem for simply connected Riemann surfaces relies on establishing a biholomorphic equivalence to one of the three model domains: the Riemann sphere $\mathbb{C}P^1$, the complex plane $\mathbb{C}$, or the unit disk $\mathbb{D}$. For the general case, the universal covering space of the Riemann surface is uniformized in this manner, and the deck transformations of the covering map determine the fundamental group action that quotients the model space to the surface. One classical approach to the proof involves solving a uniformizing partial differential equation, such as the Beltrami equation for quasiconformal mappings, combined with the Riemann mapping theorem for simply connected domains; alternatively, for specific cases like the torus, modular functions provide an explicit uniformization via elliptic functions.[](https://people.dm.unipi.it/benedett/AbikoffUNIF.pdf)[](https://www.math.mcgill.ca/gantumur/math580f11/downloads/uniformisation.pdf) Key implications of the theorem include the fact that every Riemannian metric on a torus is conformally equivalent to a flat Euclidean metric, allowing the torus to be realized as $\mathbb{C}/\Lambda$ for some lattice $\Lambda$, while for higher-genus surfaces, every metric is conformally equivalent to a hyperbolic metric of constant negative curvature, enabling representations as quotients of the hyperbolic plane by Fuchsian groups.[](https://arxiv.org/pdf/math/0505163) These conformal structures provide a geometric realization of the topological classification of surfaces, linking differential geometry with complex analysis. The models of constant curvature serve as the canonical spaces for these uniformizations.[](https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/unif7.pdf) The theorem was independently proved by [Henri Poincaré](/page/Henri_Poincaré) and [Paul Koebe](/page/Paul_Koebe) in 1907, marking a culmination of efforts dating back to [Riemann's work on conformal mappings and Fuchsian groups](/page/Riemann's_work_on_conformal_mappings_and_Fuchsian_groups). [Poincaré's proof](/page/Poincaré's_proof) appeared in *Acta Mathematica*, emphasizing the analytic continuation and monodromy aspects, while [Koebe's contributions](/page/Koebe's_contributions) in *Nachrichten der Königlichen Gesellschaft der Wissenschaften zu Göttingen* focused on the uniformization of algebraic curves and general analytic domains.[](https://eudml.org/doc/58677) ## Connections and Transport ### Riemannian Connection In differential geometry, the Riemannian connection on a surface endowed with a Riemannian metric $g$, also known as the Levi-Civita connection and denoted by $\nabla$, is the unique affine connection that preserves the metric and has vanishing torsion. This connection assigns to each pair of smooth vector fields $X$ and $Y$ on the surface a vector field $\nabla_X Y$ such that $\nabla$ is $C^\infty$-linear in both arguments and satisfies two key axioms. First, metric compatibility ensures $\nabla g = 0$, meaning X \cdot g(Y, Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z) for all smooth vector fields $X, Y, Z$.[](https://www.ime.usp.br/~gorodski/teaching/mat5771/ch2.pdf) Second, torsion-freeness requires \nabla_X Y - \nabla_Y X = [X, Y], where $[X, Y]$ is the Lie bracket of $X$ and $Y$.[](https://www.ime.usp.br/~gorodski/teaching/mat5771/ch2.pdf) These properties guarantee that the connection measures infinitesimal changes in a way that respects both the geometry induced by $g$ and the manifold's smooth structure, as originally formulated by Levi-Civita for general Riemannian manifolds, including surfaces. The Levi-Civita connection can be expressed explicitly using the Koszul formula, which derives $\nabla_X Y$ directly from the metric $g$ and the Lie brackets of vector fields: 2 g(\nabla_X Y, Z) = X \cdot g(Y, Z) + Y \cdot g(Z, X) - Z \cdot g(X, Y) + g([X, Y], Z) - g([Y, Z], X) - g([Z, X], Y) for all smooth vector fields $X, Y, Z$.[](https://www.ime.usp.br/~gorodski/teaching/mat5771/ch2.pdf) This formula proves the existence and uniqueness of the connection by showing that the right-hand side is $C^\infty$-linear in $X$ and $Y$, symmetric under interchange of $X$ and $Y$ (due to torsion-freeness), and compatible with $g$.[](https://www.ime.usp.br/~gorodski/teaching/mat5771/ch2.pdf) On a surface, where the tangent bundle is two-dimensional, the Koszul formula simplifies computations in adapted coordinates, such as orthogonal frames aligned with principal directions, though it applies generally without reliance on dimension. The curvature tensor $R$ of the Levi-Civita connection quantifies the failure of second covariant derivatives to commute and is defined by R(X, Y) Z = \nabla_X (\nabla_Y Z) - \nabla_Y (\nabla_X Z) - \nabla_{[X, Y]} Z for smooth vector fields $X, Y, Z$.[](https://www.mathematik.hu-berlin.de/~wendl/pub/connections_chapter6_2.pdf) This tensor is tensorial in all arguments and antisymmetric in $X$ and $Y$. On a surface, the two-dimensionality implies severe restrictions: the curvature tensor is fully determined by the scalar [Gaussian curvature](/page/Gaussian_curvature) $K$, satisfying R(X, Y) Z = K \bigl( g(Y, Z) X - g(X, Z) Y \bigr). [](https://www.mathematik.hu-berlin.de/~wendl/pub/connections_chapter6_2.pdf) This relation underscores that $K$ encodes all intrinsic curvature information, independent of any embedding, as established by Gauss's [Theorema Egregium](/page/Theorema_Egregium).[](https://www.gutenberg.org/files/36856/36856-pdf.pdf) In local coordinates, the connection coefficients appear as [Christoffel symbols](/page/Christoffel_symbols) $\Gamma^k_{ij}$, which can be derived from the Koszul formula by specializing to a coordinate basis.[](https://www.ime.usp.br/~gorodski/teaching/mat5771/ch2.pdf) ### Covariant Derivative and Parallel Transport In the context of differential geometry of surfaces, the covariant derivative provides a means to differentiate vector fields along curves in a way that respects the intrinsic geometry of the surface, operationalizing the Riemannian connection on the tangent bundle.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) The Riemannian connection, specifically the Levi-Civita connection, ensures that this differentiation is compatible with the metric tensor, allowing for a unique torsion-free extension of the directional derivative to curved spaces.[](https://arxiv.org/pdf/1608.04986) For a vector field $V$ defined along a curve $\gamma(s)$ on a surface, parameterized by arc length $s$, the covariant derivative $\nabla_{\gamma'(s)} V$ is defined as the orthogonal projection onto the tangent plane of the ordinary derivative $dV/ds$. This projection accounts for the embedding of the surface in ambient space, effectively removing any normal component that arises due to the curve's embedding.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) In local coordinates $(u^1, u^2)$ on the surface, if $V = V^k \partial_k$ and $\gamma(s) = (u^i(s))$, the components of the covariant derivative are given by \frac{D V^k}{ds} = \frac{d V^k}{ds} + \Gamma^k_{ij} V^i \frac{d u^j}{ds}, where $\Gamma^k_{ij}$ are the [Christoffel symbols](/page/Christoffel_symbols) of the second kind, determined by the [metric tensor](/page/Metric_tensor).[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) This formula encapsulates how the change in the vector field combines the standard coordinate derivative with corrections from the surface's curvature via the connection coefficients. Parallel transport along the curve $\gamma$ is the process of extending a vector field $V$ such that its covariant derivative vanishes, $\nabla_{\gamma'(s)} V = 0$, meaning the vector is transported without any tangential acceleration relative to the surface. This transport preserves the inner product and lengths of vectors, as the [Levi-Civita connection](/page/Levi-Civita_connection) is metric-compatible, ensuring that angles and norms remain invariant along the path.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) Solving the parallel transport equation yields a unique vector field along $\gamma$ for given initial conditions, which can be expressed as a linear ordinary differential equation in coordinates using the [Christoffel symbols](/page/Christoffel_symbols). A illustrative example occurs on the unit sphere, where parallel transport reveals the effects of curvature. Consider transporting a tangent vector around a latitude circle at colatitude $u_0$; upon completing the loop, the vector undergoes a rotation by an angle of $-2\pi \cos u_0$ relative to its initial orientation, with the rotation magnitude depending on the latitude and reflecting the enclosed Gaussian curvature.[](https://math.franklin.uga.edu/sites/default/files/users/user317/ShifrinDiffGeo.pdf) For latitudes near the pole ($u_0 \approx 0$, $\cos u_0 \approx 1$), the rotation approaches $2\pi$, highlighting how closed paths on curved surfaces can lead to non-trivial shifts in vector orientation. ### Holonomy on Surfaces In the differential geometry of surfaces, holonomy describes the transformation of tangent vectors under parallel transport along closed curves. For a closed curve $\gamma$ on an oriented Riemannian surface $S$ and a tangent vector $V$ at a base point $p \in S$, the holonomy $Hol_\gamma(V)$ is the vector obtained by parallel transporting $V$ along $\gamma$ and returning to $p$. This operation yields a linear isomorphism of the tangent space $T_p S$, and the collection of all such transformations for loops based at $p$ forms the holonomy group $Hol_p(S)$ at $p$. On surfaces, which are 2-dimensional Riemannian manifolds, the local holonomy group is a closed subgroup of $SO(2)$, the special orthogonal group in two dimensions, preserving orientation and reflecting the rotational nature of infinitesimal transformations. Globally, the full holonomy group, generated by all closed loops, depends on the topology of $S$, particularly the Euler characteristic $\chi(S)$, which influences the possible rotations through the interplay of curvature and fundamental group. For non-orientable surfaces, the holonomy lies in $O(2)$. The holonomy on surfaces is intimately linked to the Gaussian curvature $K$ via the Ambrose--Singer theorem, which asserts that the Lie algebra of the holonomy group is spanned by the values of the curvature tensor (or form) along loops in the holonomy bundle. In two dimensions, the curvature is scalar, and the theorem implies that holonomies are rotations whose angles are determined by integrals of $K$ over surfaces bounded by the loops; specifically, for a simply connected domain $D$ with boundary $\gamma$, the rotation angle is $\int_D K \, dA$. This integrated curvature governs the structure of the holonomy group. Representative examples illustrate these concepts. On the flat torus, where $K \equiv 0$, all holonomies are the identity, yielding a trivial holonomy group consistent with Euclidean geometry. In contrast, the round sphere with constant positive Gaussian curvature $K = 1/R^2$ has full holonomy group $SO(2)$, where parallel transport around loops enclosing area $A$ produces rotations by angle $K A$, and the total integral $\int_S K \, dA = 4\pi = 2\pi \chi(S)$ reflects the topology with $\chi(S) = 2$. ## Global Aspects ### Embeddings and Isometries An embedding of a surface into $\mathbb{R}^3$ is a smooth injective immersion, where the differential of the map is injective at every point, ensuring that the surface is locally Euclidean and the global map is a homeomorphism onto its image.[](https://www.math.ucsd.edu/~eizadi/250A-2019/Patrick-Girardet.pdf) Local existence of such embeddings follows from Bonnet's fundamental theorem of surface theory, which states that if a first fundamental form $I$ and second fundamental form $II$ satisfy the Gauss and Codazzi-Mainardi compatibility equations, then there exists a neighborhood around any point and an isometric immersion realizing these forms into $\mathbb{R}^3$.[](https://www.cis.upenn.edu/~cis6100/gma-v2-chap20.pdf) For global existence, particularly in the $C^1$ category, the Nash-Kuiper theorem asserts that any smooth Riemannian metric on an orientable 2-manifold admits a $C^1$ isometric immersion into $\mathbb{R}^3$, allowing highly wrinkled realizations even when smooth immersions do not exist. For metrics admitting smooth isometric immersions into $\mathbb{R}^3$, these can be approximated arbitrarily closely by $C^1$ isometric immersions. Recent extensions achieve Hölder continuity $C^{1,\theta}$ for $\theta < 1/5$ in certain cases, such as for disks.[](https://arxiv.org/abs/1906.08608) An isometry between two surfaces is a diffeomorphism $\phi$ such that the pullback $\phi^* I = I$, preserving the first fundamental form and thus all intrinsic geometric quantities like lengths, angles, and Gaussian curvature via the Theorema Egregium.[](https://graphics.stanford.edu/courses/cs468-13-spring/assets/lecture15.pdf) For surfaces with strictly positive Gaussian curvature $K > 0$, the Cohn-Vossen rigidity theorem establishes that any isometric immersion into $\mathbb{R}^3$ is unique up to rigid motions of $\mathbb{R}^3$ (translations and rotations), meaning the embedding is rigid and cannot be deformed while preserving the metric.[](https://math.jhu.edu/~js/rigidity.pdf) The Weyl problem addresses the global uniqueness of embeddings for metrics on the sphere: given a smooth metric on $S^2$ with positive [Gaussian curvature](/page/Gaussian_curvature) $K > 0$, there exists a unique [convex](/page/Convex) [isometric](/page/Isometric) [embedding](/page/Embedding) into $\mathbb{R}^3$ up to isometries of $\mathbb{R}^3$, as solved affirmatively by Nirenberg for the analytic case and extended to smooth metrics by subsequent works. Extensions to nonnegative curvature $K \geq 0$ maintain uniqueness under additional smoothness and boundary conditions, ensuring the embedding remains [convex](/page/Convex). ### Topology of Surfaces The topology of compact surfaces is fundamentally determined by two invariants: orientability and genus. Orientable compact surfaces without boundary are classified up to [homeomorphism](/page/Homeomorphism) by their genus $g$, a non-negative integer representing the number of "handles" or tori in their connected sum decomposition. The sphere corresponds to genus 0, the [torus](/page/Torus) to genus 1, and surfaces of genus $g \geq 2$ to the connected sum of $g$ tori.[](https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) Non-orientable compact surfaces are classified similarly by the number of crosscaps, with the real [projective plane](/page/Projective_plane) ($\mathbb{RP}^2$) as the basic non-orientable surface (equivalent to one crosscap), and higher cases as connected sums thereof, such as the [Klein bottle](/page/Klein_bottle) for two crosscaps.[](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Huang.pdf) This classification theorem asserts that every compact connected surface is homeomorphic to either a sphere, a connected sum of tori, or a connected sum of projective planes.[](https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) A key topological invariant is the Euler characteristic $\chi$, which distinguishes surfaces within these classes. For closed orientable surfaces of genus $g$, $\chi = 2 - 2g$; thus, the sphere has $\chi = 2$, the torus $\chi = 0$, and higher-genus surfaces have negative even integers.[](https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) For non-orientable surfaces with $k$ crosscaps, $\chi = 2 - k$, yielding $\chi = 1$ for the projective plane and $\chi = 0$ for the Klein bottle.[](https://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Huang.pdf) The Euler characteristic is computed from a cell complex decomposition as $\chi = v - e + f$, where $v$, $e$, and $f$ are the numbers of vertices, edges, and faces, and it remains invariant under homeomorphisms.[](https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) The [fundamental group](/page/Fundamental_group) $\pi_1$ provides another classifying invariant, capturing the homotopy classes of loops. For the orientable surface of [genus](/page/Genus) $g$, $\pi_1$ has [presentation](/page/Presentation) $\langle a_1, b_1, \dots, a_g, b_g \mid \prod_{i=1}^g [a_i, b_i] = 1 \rangle$, where $[a_i, b_i] = a_i b_i a_i^{-1} b_i^{-1}$; this group is free abelian of rank 2 for $g=1$ (the [torus](/page/Torus)) and non-abelian for $g \geq 2$.[](https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) For the sphere ($g=0$), $\pi_1$ is trivial. For non-orientable surfaces with $k$ crosscaps, $\pi_1$ has presentation $\langle a_1, \dots, a_k \mid a_1^2 \cdots a_k^2 = 1 \rangle$, which is infinite for $k \geq 2$ and $\mathbb{Z}/2\mathbb{Z}$ for the [projective plane](/page/Projective_plane).[](https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) These groups distinguish surfaces: distinct genera yield non-isomorphic fundamental groups. Covering spaces further illuminate the topology, with the universal cover being the simply connected space that covers the surface. [The sphere](/page/The_Sphere) is its own universal cover. For the torus, the universal cover is the [Euclidean plane](/page/Euclidean_plane) $\mathbb{R}^2$. For orientable surfaces of [genus](/page/Genus) $g \geq 2$, the universal cover is the hyperbolic plane $\mathbb{H}^2$. Non-orientable surfaces admit an orientable double cover whose universal cover aligns with the above cases based on the corresponding orientable [genus](/page/Genus). These universal covers relate to conformal structures via the [uniformization theorem](/page/Uniformization_theorem), which models surfaces as quotients of [the sphere](/page/The_Sphere), plane, or hyperbolic plane by discrete groups of isometries.[](https://pi.math.cornell.edu/~hatcher/AT/AT.pdf) ### Global Curvature Integrals Global curvature integrals in differential geometry extend local notions of curvature to integral quantities over entire surfaces, providing profound links between geometry and topology. A central example is the total Gaussian curvature of a compact, oriented surface $S$ without boundary, given by $\int_S K \, dA = 2\pi \chi(S)$, where $K$ is the Gaussian curvature and $\chi(S)$ is the Euler characteristic of $S$.[](https://www2.math.upenn.edu/~shiydong/Math501X-7-GaussBonnet.pdf) This equality, a direct application of the Gauss-Bonnet theorem, implies that the integrated curvature depends solely on the surface's topology, independent of its specific embedding in $\mathbb{R}^3$. For instance, all genus-zero surfaces like the sphere have total curvature $4\pi$, while tori yield $0$.[](http://math.uchicago.edu/~may/REU2015/REUPapers/Butt.pdf) Another significant global integral is the Willmore energy, defined for an immersed surface $\Sigma \subset \mathbb{R}^3$ as $W(\Sigma) = \int_\Sigma H^2 \, d\mu$, where $H$ is the mean curvature and $d\mu$ is the area element. This energy measures the surface's deviation from sphericity and is conformally invariant. Among all closed immersed surfaces, round spheres achieve the global minimum $W(\Sigma) = 4\pi$, with equality holding uniquely for spheres up to congruence.[](http://www.math.uchicago.edu/~aneves/papers/Wsurv) The Willmore energy has been extensively studied, with its minimizers providing insights into optimal embeddings and variational problems in geometry.[](https://arxiv.org/abs/1202.6036) These integrals find applications in discrete settings, such as polyhedral surfaces, where [Gaussian curvature](/page/Gaussian_curvature) concentrates at vertices as angular defects. For a closed polyhedral surface, the total curvature is the sum of these defects over all vertices, equaling $2\pi \chi(S)$, mirroring the smooth case.[](https://www2.math.upenn.edu/~shiydong/Math501X-7-GaussBonnet.pdf) Additionally, global curvature integrals guide approximation methods via [curvature flows](/page/Curvature); for example, the Willmore flow, the $L^2$-gradient flow of the Willmore [energy](/page/Energy), evolves surfaces toward energy minimizers like spheres, useful in smoothing polyhedral approximations or modeling biological membranes.[](http://www.math.uchicago.edu/~aneves/papers/Wsurv) For non-compact surfaces, extensions of the Gauss-Bonnet theorem require growth conditions on the curvature to ensure convergence of the integrals. Specifically, for complete, oriented non-compact surfaces with finite total Gaussian curvature, the integral $\int_S K \, dA \leq 2\pi \chi(S)$, with equality under additional compactness-like conditions at infinity, such as those in the Cohn-Vossen theorem.[](https://www.ams.org/proc/1982-086-01/S0002-9939-1982-0663893-8/S0002-9939-1982-0663893-8.pdf) These results highlight how global integrals capture asymptotic behavior and topological structure in unbounded domains.

References

  1. [1]
    [PDF] Differential Geometry of Surfaces - People @EECS
    Differential geometry of a 2D manifold or surface embedded in 3D is the study of the intrinsic properties of the surface as well as the ef-.
  2. [2]
    [PDF] Basics of the Differential Geometry of Surfaces - CIS UPenn
    The purpose of this chapter is to introduce the reader to some elementary concepts of the differential geometry of surfaces. Our goal is rather modest: We ...
  3. [3]
    [PDF] DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces
    An Introduction to Hyperbolic Geometry 91. 3. Surface Theory with Differential Forms 101. 4. Calculus of Variations and Surfaces of Constant Mean Curvature 107.
  4. [4]
    [PDF] 1 The Differential Geometry of Surfaces
    The concept of a manifold provides us with a general notion of a surface. For dealing with surfaces that bound 3-dimensional bodies, we will want to add some ...
  5. [5]
    [PDF] General investigations of curved surfaces of 1827 and 1825 ...
    Gauss's Paper of 1827, General Investigations of Curved Surfaces ... 1 ... surfaces and their results cover asignificant portion of the domain of higher geometry,.Missing: history | Show results with:history
  6. [6]
    Outline of a History of Differential Geometry (II)
    GAUSS (1777-I855). For us the activity of GAUSS is threefold: as inventor of non- Euclidean geometry, as inventor of intrinsic differential geometry, and as a ...
  7. [7]
    [PDF] Differential Geometry of Curves and Surfaces
    In this course we will deal with curves living in the plane and in three-dimensional space as well as with surfaces living in three-dimensional space. A curve ...
  8. [8]
    [PDF] cs 468 notes: differential geometry for computer science - Arun Debray
    A more general application to point clouds is Poisson representation, which solves a PDE on the volume around a point set to obtain the normals. This creates a ...
  9. [9]
    [PDF] differential-geometry-2024.pdf - Harvard Mathematics Department
    General relativity studies solutions of these equations as they tell how matter bends space. The geodesic equations then tell, how matter moves in this space.Missing: architecture | Show results with:architecture
  10. [10]
    [PDF] Geometry of Architectural Freeform Structures
    Nov 26, 2008 · It is a well known theorem of classical differential geometry that the following properties of a surface are essentially equivalent: (i) the ...
  11. [11]
    Leonhard Euler (1707 - 1783) - Biography - MacTutor
    Euler made substantial contributions to differential geometry, investigating the theory of surfaces and curvature of surfaces. Many unpublished results by ...
  12. [12]
    Gaspard Monge - Biography - MacTutor - University of St Andrews
    Gaspard Monge is considered the father of differential geometry because of his work Application de l'analyse à la géométrie where he introduced the concept ...Missing: 1700s | Show results with:1700s
  13. [13]
    Carl Friedrich Gauss - Biography
    ### Summary of Gauss's Work on 1827 Disquisitiones Generales circa superficies curvas, Theorema Egregium, and Intrinsic Geometry
  14. [14]
    Gaston Darboux (1842 - 1917) - Biography - MacTutor
    Gaston Darboux was the son of François Darboux (1800-1849) and Alix Gourdoux (1811-1887). François was a clothes merchant and haberdasher.Missing: global 19th
  15. [15]
    Henri Poincaré - Biography
    ### Summary of Poincaré's Contributions to the Uniformization Theorem and Connections to Riemann Surfaces
  16. [16]
    [PDF] arXiv:2008.12189v2 [math.CV] 3 Sep 2021
    Sep 3, 2021 · Paul Koebe and shortly thereafter Henri Poincaré are credited with having proved in 1907 the famous uniformization theorem for Riemann surfaces ...Missing: early | Show results with:early
  17. [17]
    General Relativity at 100 | - American Mathematical Society
    Dec 2, 2015 · On November 25, 1915, Einstein's paper on general relativity, Die Feldgleichungen der Gravitation (The Field Equations of Gravity), ...
  18. [18]
    [PDF] A History of Curves and Surfaces in CAGD - FarinHansford.com
    Introduction. The term CAGD was coined by R. Barnhill and R. Riesenfeld in 1974 when they organized a conference on that topic at the University of Utah.
  19. [19]
    [PDF] Surfaces
    Feb 13, 2014 · These notes summarize the key points in the second chapter of Differential. Geometry of Curves and Surfaces by Manfredo P. do Carmo.
  20. [20]
    [PDF] Differential Geometry: - Surfaces and Parameterizations
    Definition: A subset S⊂R3, is a regular surface if for every p∈S there exists a neighborhood V⊂R3, and a map Φ:U→V∩S of an open set U⊂R2 onto V∩S such that:.
  21. [21]
    [PDF] lee-smooth-manifolds.pdf - MIT Mathematics
    TAKEUTIIZARING. Introduction to. 34 SPITZER. Principles of Random Walk. Axiomatic Set Theory. 2nd ed. 2nd ed. 2 OXTOBY. Measure and Category. 2nd ed.
  22. [22]
    [PDF] Differential Geometry - Lecture Notes - UC Berkeley math
    Maps φ as in the definition are called charts for X. A collection of charts whose domains cover X is called an atlas for X. If φα : Uα → Vα,φβ : Uβ → Vβ ...
  23. [23]
    [PDF] General Investigations of Curved Surfaces - Project Gutenberg
    In 1827 Gauss presented to the Royal Society of Göttingen his important paper on the theory of surfaces, which seventy-three years afterward the eminent ...Missing: Theorema Egregium
  24. [24]
    None
    Below is a merged summary of the tangent space and related concepts from "Differential Geometry of Curves and Surfaces" by Manfredo P. do Carmo, consolidating all information from the provided segments into a single, comprehensive response. To maximize detail and clarity, I will use a table format in CSV style for key concepts, followed by additional narrative details where necessary. The response retains all information mentioned across the summaries, organized by topic, with page references and additional notes.
  25. [25]
    [PDF] M462 (HANDOUT 9) 0.1. Christoffel symbols. Let S be a regular ...
    To determine the Christoffel symbols, we take inner products of the above equa- tions with Su etc. <Suu,Su> = Γ1. 11E + Γ2. 11F. On the other hand, ...
  26. [26]
    [PDF] Elementary Differential Geometry of Surfaces - Dr. Wolfgang Lindner
    The Gauss equations are a system of three partial differential equations. ... The Codazzi-Mainardi equations are important formulas which link together the metric.
  27. [27]
    [PDF] Differential Geometry: a concise introduction - UC Homepages
    2.4 Gauss-Weingarten & Gauss-Codazzi equations. 36. Problem 15. Prove: for a curvature line parametrization the Codazzi equation(s) reads. 0 = κ1v + Ev. 2E (κ1 ...
  28. [28]
    [PDF] DIFFERENTIAL GEOMETRY OF CURVES AND SURFACES 6 ...
    Gauss' Theorema Egregium. Question. How can we decide if two given surfaces can be obtained from each other by. “bending without stretching”? The simplest ...
  29. [29]
    [PDF] Gauss' Theorema Egregium
    Mar 2, 2017 · The idea is to show that K can be expressed purely in terms of E, F, G and their 61st and 2nd) partials.Missing: statement | Show results with:statement
  30. [30]
    [PDF] curvature and the theorema egregium of gauss
    Dec 31, 2015 · In this note, we describe a simple way to define the second fundamental form of a hypersur- face in Rn and use it to prove Gauss's Theorema ...
  31. [31]
    [PDF] Classifying Quadrics using Exact Arithmetic - Geometric Tools
    Oct 10, 2022 · Classify quadrics by using exact arithmetic with rational coefficients to classify the solution set, which can be ellipsoids, hyperboloids, ...<|separator|>
  32. [32]
    [PDF] Surface Curves and Fundamental Forms∗
    Dec 3, 2024 · The first fundamental form will change when the surface patch is changed. Example 1. For the unit sphere parametrized with latitude and ...<|control11|><|separator|>
  33. [33]
    Ellipsoid gaussian curvature - Applied Mathematics Consulting
    Oct 7, 2019 · For an ellipsoid with equation. \left(\frac{x}{a}\right^2 +. the Gaussian curvature at each point is given by. K(x,y,z) = \frac{1}{a.
  34. [34]
    [PDF] Gaussian and Mean Curvatures∗
    The change rate of n in a tangent direction, i.e., the normal curvature, indicates the degree of variation of surface geometry in that direction at the point.
  35. [35]
    [PDF] The classical theory of minimal surfaces
    We present here a survey of recent spectacular successes in classical minimal surface theory. We highlight this article with the theorem that the plane, the.Missing: seminal | Show results with:seminal
  36. [36]
    [PDF] Minimal surfaces for undergraduates - arXiv
    Jan 7, 2021 · In this section we derive Lagrange's equation of minimal graphs, which is one of the first examples in the calculus of variations for functions ...
  37. [37]
    [PDF] Minimal Surfaces
    Dec 13, 2012 · In 1864, Alfred Enneper discovered a minimal surface conjugate to itself, now called the Enneper Surface. Below is the surface, together with ...
  38. [38]
    [PDF] Ruled surfaces and developable surfaces
    Folklore Statement 1.11 Developable surfaces can be decom- posed into planar parts, cylinders, cones, and tangent surfaces. (which are swept by the tangents of ...
  39. [39]
    [PDF] Geodesic curvature
    T,n,N are positively oriented. Then α. //. = knN + kg n. kg is called the geodesic curvature of α in M (with respect to the orientation N). Page 2. Geodesic ...
  40. [40]
    [PDF] Classical Differential Geometry Peter Petersen
    We start with a characterization in terms of Gauss curvature. Lemma 5.5.6. (Monge, 1775) A surface with vanishing Gauss curvature and no umbilics is a ...
  41. [41]
    [PDF] Unit 12: The exponential map
    On U there are coordinates (ρ, θ) such that g = I = 1 0. 0 G satisfying limρ→0 G(ρ, θ)=1. Proof. These are called geodesic polar coordinates because they come ...
  42. [42]
    https://www.worldscientific.com/doi/pdf/10.1142/9789812812834_0033
  43. [43]
    [PDF] 7. THE GAUSS-BONNET THEOREM - Penn Math
    π = T . The sides of the triangle are geodesics, that is, arcs of great circles. Extend these arcs to full great circles,.<|control11|><|separator|>
  44. [44]
    [PDF] Gauss-Bonnet Theorem - UChicago Math
    Jul 30, 2010 · Though this paper presents no original mathematics, it carefully works through the necessary tools for proving Gauss-Bonnet. Gauss first proved ...
  45. [45]
    [PDF] From Foucault's Pendulum to the Gauss–Bonnet Theorem - arXiv
    Nov 5, 2017 · Given the right intuition about geodesics and parallel transport, one can prove the Gauss-Bonnet theorem for embedded surfaces with little more ...
  46. [46]
    The Pseudosphere
    A theorem first proved by David Hilbert states that it is impossible to embed the entire hyperbolic plane isometrically as a surface in Euclidean space. On the ...
  47. [47]
    [PDF] hilbert's theorem on immersion of the hyperbolic plane
    Aug 29, 2020 · Theorem 4.1 (Hilbert). There exists no isometric immersion of a complete surface with constant negative Gaussian curvature in R3. Hilbert's ...
  48. [48]
    Ueber die Uniformisierung beliebiger analytischer Kurven - EuDML
    Koebe, P.. "Ueber die Uniformisierung beliebiger analytischer Kurven." Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch ...Missing: DOI | Show results with:DOI
  49. [49]
    [PDF] Curvature and Uniformization - Michael Taylor
    Abstract. We approach the problem of uniformization of general Riemann sur- faces through consideration of the curvature equation, and in particular.Missing: citation | Show results with:citation
  50. [50]
    [PDF] The Uniformization Theorem Author(s): William Abikoff Source - unipi
    The analytic configurations are Riemann surfaces which lie in the sheaf of germs of holomorphic or meromorphic functions on C. Overlapping discs define the ...
  51. [51]
    [PDF] The Uniformisation Theorem
    Dec 12, 2011 · The proof outline is as follows: (1) Define harmonic and subharmonic functions on Riemann surfaces. (2) Define a Perron family of subharmonic ...
  52. [52]
    [PDF] A note on uniformization of Riemann surfaces by Ricci flow - arXiv
    In this note, we clarify that the Ricci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.<|control11|><|separator|>
  53. [53]
    Ueber die Uniformisierung reeller algebraischer Kurven - EuDML
    Koebe, P.. "Ueber die Uniformisierung reeller algebraischer Kurven." Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch ...Missing: DOI | Show results with:DOI
  54. [54]
    [PDF] Connections - IME-USP
    Let M be a smooth manifold. A (Koszul) connection in M is a bilinear map ∇ : Γ(TM)×Γ(TM) → Γ(TM), where we write ∇XY instead of ∇(X, Y ), such that a. ∇fX Y = ...
  55. [55]
    [PDF] Chapter 6 Curvature in Riemannian Geometry
    The function KG : Σ → R is called the Gaussian curvature, and despite appearances to the contrary, we will find that it does not depend on the embedding Σ ֒→ R3 ...
  56. [56]
    [PDF] On the history of Levi-Civita's parallel transport - arXiv
    Aug 6, 2016 · case, therefore, we strictly follow first the original paper of Levi-Civita, i.e., [35], hence other ... called the Levi-Civita's connection, on ...
  57. [57]
    [PDF] 3 Immersions and Embeddings - UCSD Math
    This example shows the importance of the immersion condition as part of the definition of a smooth embedding - the image of α in R2 is a curve with a cusp at.
  58. [58]
    [1906.08608] Global Nash-Kuiper theorem for compact manifolds
    Jun 20, 2019 · In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity ...
  59. [59]
    [PDF] CS 468 (Spring 2013) — Discrete Differential Geometry
    Isometries and local isometries. • Definition of isometric surfaces: two surfaces M and N are isometric if there exists a mapping φ : M → N so that hDφ(Xp), Dφ ...
  60. [60]
    [PDF] rigidity of nonnegatively curved surfaces relative to a curve
    May 15, 2018 · We prove that any properly oriented C2,1 isometric immersion of a positively curved Riemannian surface M into Euclidean 3-space is uniquely de-.
  61. [61]
    [PDF] Algebraic Topology - Cornell Mathematics
    This book was written to be a readable introduction to algebraic topology with rather broad coverage of the subject. The viewpoint is quite classical in ...
  62. [62]
    [PDF] classification of surfaces - UChicago Math
    Abstract. We will classify compact, connected surfaces into three classes: the sphere, the connected sum of tori, and the connected sum of projective.
  63. [63]
    [PDF] the gauss-bonnet theorem - UChicago Math
    The Gauss-Bonnet theorem is perhaps one of the deepest theorems of differential geometry. It relates a compact surface's total Gaussian curvature to its Euler.
  64. [64]
    [PDF] The Willmore Conjecture - UChicago Math
    In this survey article we discuss the history and our recent solution of the Willmore conjecture, the problem of determining the best torus among all. We begin ...
  65. [65]
    [1202.6036] Min-Max theory and the Willmore conjecture - arXiv
    Feb 27, 2012 · In 1965, TJ Willmore conjectured that the integral of the square of the mean curvature of a torus immersed in Euclidean three-space is at least 2\pi^2.Missing: original | Show results with:original
  66. [66]
    Gauss-Bonnet theorems for noncompact surfaces
    The aim of this note is to give short proofs of the following two theorems, due to. Cohn-Vossen [3] and Huber [4] respectively. Theorem A (Gauss-Bonnet ...