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Differential geometry

Differential geometry is the study of geometric structures using and , providing a mathematical framework for understanding curves, surfaces, and higher-dimensional manifolds. It focuses on the properties of these objects through techniques from and integral , enabling the analysis of how shapes bend, curve, and interact in various spaces. At its core, differential geometry intersects , , and classical , offering tools to describe smooth objects like differentiable manifolds—generalizations of curves and surfaces to arbitrary dimensions. Central concepts include the at a point on a manifold, which localizes linear approximations, and tensor fields that generalize vectors and scalars to capture multilinear relationships. emerges as a fundamental invariant, quantifying deviations from flatness, while geodesics represent the "straightest" paths analogous to lines in flat space. Riemannian metrics further define inner products on tangent spaces, allowing measurements of lengths, angles, and volumes on curved manifolds. The field emphasizes both local and global properties: locally, it examines behaviors via and Jacobians; globally, it explores and through and . Classical differential geometry, originating in the with works on curves and surfaces by Gauss and others, has evolved into modern branches like for and for algebraic varieties. Applications of differential geometry span physics, , and , underpinning Einstein's by modeling as a with variable dictated by mass and energy. In applied contexts, it informs for stress on deformed bodies, computer tomography for reconstructing images from projections, and for optimizing motion on non-Euclidean terrains. Its rigorous yet concrete approach continues to drive advancements in theories and geometric .

History

Ancient origins to Renaissance

The foundational concepts of differential geometry trace their roots to mathematics, where geometric reasoning laid the groundwork for understanding curved spaces without analytic tools. Euclid's Elements, composed around 300 BC, established the axioms of plane geometry, including postulates on straight lines, circles, and parallel lines that formed the basis for deductive proofs of spatial relationships. These axioms emphasized the properties of flat Euclidean space, providing a rigorous framework that influenced later explorations of curvature by treating geometry as a system of propositions derived from primitive notions. Euclid's work systematized earlier Greek contributions, such as those from Thales and Pythagoras, into a cohesive structure that prioritized logical consistency over empirical measurement. Building on this, advanced the study of curved figures in the through his , a precursor to that approximated areas and volumes of non-polygonal shapes by inscribing and circumscribing polygons. In treatises like , applied this method to compute the surface area and volume of spheres and paraboloids, demonstrating how curved surfaces could be quantified via limits of polygonal approximations without invoking infinitesimals. His approach highlighted the distinction between straight-edged and curved forms, using mechanical principles like the to verify geometric results, thus bridging with physical intuition. A key development in the study of curves came from (c. 262–190 BC), whose eight-volume Conics provided a systematic treatment of conic sections—parabolas, ellipses, and hyperbolas—derived from plane sections of cones, introducing terminology and properties that influenced later algebraic and differential treatments of curves. Similarly, Ptolemy's Geography in the 2nd century AD introduced early notions of in by devising map projections that accounted for the Earth's spherical shape, such as conic projections that preserved angles while distorting distances to represent global features on a plane. During the , scholars expanded these ideas through algebraic and geometric innovations. , in the 9th century, contributed to in his Book of Algebra and Almucabala, classifying quadratic equations geometrically to provide tools for analyzing non-linear spatial forms. , in the , further developed by solving cubic equations through intersections of conic sections, as detailed in his Treatise on the Demonstration of Problems of Algebra, offering visual methods to resolve higher-degree polynomials that anticipated coordinate-based approaches. In the , these ancient and medieval legacies were revitalized, with figures like integrating geometric insights into artistic and engineering sketches. Da Vinci's 15th-century notebooks contain detailed drawings of curved surfaces, such as anatomical forms and mechanical devices, where he explored to depict three-dimensional on two-dimensional planes, emphasizing optical distortions and shadows to convey depth. The rediscovery of Ptolemy's Geography in the , translated into Latin around 1406, spurred renewed interest in map projections during this period, influencing cartographers like to refine methods for representing spherical on flat maps. These efforts set the stage for the integration of in subsequent centuries, transitioning geometric intuitions toward analytic precision.

Post-calculus developments

The development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz in the late 17th century provided the foundational tools for parametrizing curves, allowing mathematicians to analyze their geometric properties dynamically through derivatives and integrals. Newton's fluxions and Leibniz's differentials enabled the representation of curves as functions of a parameter, shifting focus from static Euclidean constructions to the study of rates of change, such as tangents and arc lengths, which laid the groundwork for differential geometry. In the 1740s, Leonhard Euler advanced this framework by applying to curves and surfaces of revolution, exploring their generation through rotation and deriving properties like via differential equations. Euler's investigations included parametrizing such surfaces and examining their intrinsic features, in addition to his earlier polyhedral formula V - E + F = 2 (published in 1758 but conceived in the 1750s). This work emphasized the analytical treatment of surfaces as loci defined by revolving curves, introducing methods to compute volumes and areas that influenced subsequent surface theory. Gaspard Monge, in the late , further integrated with through his development of descriptive geometry, a system for projecting three-dimensional objects onto planes to facilitate and architectural design. Published in his 1799 lectures Géométrie descriptive, this approach used orthogonal projections to represent surfaces accurately, enabling the solution of spatial problems via planar drawings. Concurrently, Monge pioneered the use of partial differential equations to describe surfaces, particularly in his 1785 memoir on curved surface generation, where he linked geometric loci to first-order PDEs like those governing normals and developable surfaces. His Application de l'analyse à la géométrie (1795) introduced lines of curvature as integral curves of principal directions, providing a differential framework for surface analysis. Carl Friedrich Gauss contributed early insights to surface theory in 1827, rooted in his analytical pursuits, by conceptualizing the Gauss map, which associates each point on a surface to the unit sphere via its normal vector, quantifying how the surface bends in space. This mapping, formalized in his Disquisitiones generales circa superficies curvas, allowed for the measurement of total curvature and foreshadowed extrinsic differential invariants. Gauss's approach treated surfaces as parametrized by coordinates, using partial derivatives to define metric properties and deviations from flatness. A key outcome of these extrinsic analyses was the Frenet-Serret formulas for space curves, which describe the evolution of the \mathbf{T}, \mathbf{N}, and binormal \mathbf{B} frames along the s, incorporating \kappa and torsion \tau as functions of s: \begin{align*} \frac{d\mathbf{T}}{ds} &= \kappa \mathbf{N}, \\ \frac{d\mathbf{N}}{ds} &= -\kappa \mathbf{T} + \tau \mathbf{B}, \\ \frac{d\mathbf{B}}{ds} &= -\tau \mathbf{N}. \end{align*} Originally derived by Joseph Serret in 1851 and independently by Jean-Frédéric Frenet in 1852, these equations encapsulated the twisting and bending of curves in three dimensions, building directly on 18th-century parametrization techniques. Torsion \tau quantifies deviation from planarity, while \kappa measures local bending, providing differential invariants essential for understanding space curve geometry.

Rise of intrinsic and non-Euclidean geometry

In the early 19th century, made a pivotal advancement in the understanding of curved surfaces with his , published in 1827 as part of his Disquisitiones generales circa superficies curvas. This theorem demonstrated that the of a surface is an intrinsic property, determined solely by measurements within the surface itself, independent of its in three-dimensional . Gauss's insight shifted focus from extrinsic descriptions—relying on external coordinates—to intrinsic geometry, where properties like could be computed using distances and angles measured by inhabitants of the surface. Parallel to these developments, the foundations of emerged through the independent efforts of and in the 1820s and 1830s. , working in , developed a complete system of by 1823, rejecting Euclid's and allowing multiple parallels through a point to a given line; his work appeared in print in 1832 as an appendix to his father's book on geometry. Similarly, in published the first account of in 1829 in the Kazan Messenger, constructing a consistent axiomatic framework where the sum of angles in a is less than 180 degrees, leading to spaces of constant negative curvature. These innovations challenged the universality of and highlighted the possibility of alternative geometries with different parallel properties. Building on these ideas, delivered his lecture in 1854 at the , titled Über die Hypothesen, welche der Geometrie zu Grunde liegen, which introduced the concept of n-dimensional manifolds equipped with a . Riemann generalized intrinsic to abstract spaces of arbitrary dimension, defining them through a quadratic that measures distances locally, thus encompassing both and non-Euclidean cases depending on the sign. His framework allowed for positive, zero, or negative in higher dimensions, providing a unified language for spaces where is determined by the rather than rigid . To demonstrate the consistency of hyperbolic geometry, Eugenio Beltrami provided the first concrete models in the late 1860s. In his 1868 paper Saggio di interpretazione della geometria non-euclidea, Beltrami constructed a model within using the , showing that hyperbolic axioms could be realized without contradiction, and extended this to higher dimensions in 1869. further refined these ideas in the 1870s, particularly in his 1871 work Über die sogenannte nicht-euklidische Geometrie, developing the projective model (now known as the Beltrami-Klein model) inside a disk, where straight lines are chords and the metric is induced projectively to preserve hyperbolic properties. These models embedded in Euclidean settings, confirming its validity and facilitating computations. A central concept in these intrinsic geometries is the , defined as the shortest path between two points on a manifold, analogous to straight lines in but curved according to the . In the spherical plane, with positive , geodesics are great circles, such as the or meridians on a , where the distance between points exceeds the Euclidean straight-line measure due to the surface's convexity. In contrast, the hyperbolic plane exhibits negative , where geodesics diverge more rapidly, allowing triangles to have angle sums less than 180 degrees; for example, in Beltrami's model, these paths follow rulings that spread apart, illustrating in area. These examples underscore how intrinsic dictate global structure, paving the way for extensions to higher-dimensional Riemannian spaces in subsequent developments.

Modern era and contemporary advances

In the 20th century, differential geometry evolved from its 19th-century foundations in intrinsic and non-Euclidean geometries into a rigorous framework integrating , , and physics, with profound advancements in global invariants and geometric evolution equations. extended in the 1920s and 1930s through his method of moving frames, which provides a coordinate-free approach to describing the geometry of manifolds using orthonormal frames adapted to the local structure. This generalization allows for the study of non-holonomic spaces and connections beyond metrics, influencing modern treatments of and symmetry groups. Cartan's framework synthesizes theory with differential forms, enabling the formulation of in terms of frame derivatives. In the 1940s, Shiing-Shen Chern introduced characteristic classes as global invariants of fiber bundles, expressed via differential forms on the base manifold. These classes, now known as Chern classes, capture topological obstructions to sections and flat connections, with applications to the Gauss-Bonnet theorem in higher dimensions. Chern's construction, developed independently of André Weil, uses the curvature of a connection to define cohomology classes that are independent of the choice of connection. This work revitalized global differential geometry and laid groundwork for modern gauge theory. The Atiyah-Singer index theorem, proved in 1963 by and , connects the analytic index of elliptic operators on a compact manifold to topological invariants via characteristic classes. It states that for a Dirac-type operator, the index equals an integral of the A-hat genus times the Todd class of the bundle, linking local differential geometry to global topology. This theorem unifies previous results like the Hirzebruch-Riemann-Roch theorem and has far-reaching implications in and spectral geometry. A landmark achievement in the early 21st century was Grigory Perelman's 2002-2003 proof of the using , a parabolic that deforms a Riemannian to reduce irregularities. Perelman introduced entropy functionals and techniques to handle singularities, showing that any simply connected closed evolves to a under this flow, thus confirming as well. His preprints on provided the core arguments, verified through extensive checks by the mathematical community. Post-2000 developments have extended geometric flows into interdisciplinary applications, notably in and . In , Ricci flow interprets the training dynamics of deep s as metric evolution on data manifolds, quantifying geometric changes across layers to improve optimization and . For instance, recent frameworks model transformations as discrete Ricci flows, revealing curvature-driven feature extraction. In , geometric flows arise from unified frameworks, evolving complex structures like Calabi-Yau manifolds to satisfy conditions, with applications to moduli stabilization and entropy. These flows, often involving (2,0)-forms in almost-complex geometries, bridge differential geometry with .

Foundations

Smooth manifolds and charts

A smooth manifold is a second-countable Hausdorff topological space M that is locally Euclidean, meaning every point in M has an open neighborhood homeomorphic to an open of \mathbb{R}^n for some fixed n \geq 0. This local resemblance to space provides the foundation for defining differentiable structures globally. To equip M with a smooth (i.e., C^\infty) structure, one selects an atlas: a collection of charts (U_\alpha, \phi_\alpha), where each U_\alpha is an open of M, each \phi_\alpha: U_\alpha \to \mathbb{R}^n is a homeomorphism onto its image, and the transition maps \phi_\beta \circ \phi_\alpha^{-1}: \phi_\alpha(U_\alpha \cap U_\beta) \to \phi_\beta(U_\alpha \cap U_\beta) are C^\infty diffeomorphisms whenever U_\alpha \cap U_\beta \neq \emptyset. Two atlases are compatible if their union is also an atlas, and a maximal atlas (differential structure) is an equivalence class of compatible atlases. This ensures that differentiability is well-defined independently of chart choices. The dimension n of the manifold is invariant: if overlapping charts have different dimensions, the space cannot be locally Euclidean in a consistent way, as established by the invariance of domain theorem applied locally. A smooth manifold is orientable if it admits an oriented atlas, where all transition maps have positive Jacobian determinants (i.e., preserve orientation). Non-orientable examples include the real projective plane \mathbb{RP}^2, constructed as the quotient of the sphere S^2 by identifying antipodal points, where transition maps reverse orientation in certain charts. Orientability captures whether a consistent "handedness" can be assigned across the entire space. Classic examples illustrate these concepts. The \mathbb{R}^n itself forms a smooth manifold of n with the identity map as its single global . The n-sphere S^n = \{x \in \mathbb{R}^{n+1} \mid \|x\| = 1\} is a compact orientable manifold of n, covered by charts via from the north and south poles, yielding smooth transition maps. The n-torus T^n = S^1 \times \cdots \times S^1 (product of n circles) is a compact orientable manifold of n, inheriting its smooth structure from the product atlas of the circle's charts. The \mathbb{RP}^n = S^n / \sim (antipodal identification) is a compact manifold of n, non-orientable for even n, with charts using affine patches excluding a coordinate . Smooth manifolds possess the partition of unity property: for any open cover \{U_\alpha\} of M, there exists a subordinate partition of unity \{\rho_\alpha\}, a collection of smooth functions \rho_\alpha: M \to [0,1] with \operatorname{supp}(\rho_\alpha) \subset U_\alpha, locally finite, and \sum \rho_\alpha = 1. This holds because second-countable Hausdorff manifolds are paracompact, admitting locally finite refinements of any open cover, and paracompact Hausdorff spaces support such partitions. Partitions of unity enable gluing local constructions into global ones, essential for defining geometric objects like metrics or embeddings. A fundamental result is the : every smooth n-dimensional manifold admits a smooth into \mathbb{R}^{2n}, realizing it as a closed . This theorem underscores that smooth manifolds can be studied within the familiar framework of , facilitating the transfer of analytic tools. The also allows defining spaces at each point as the of derivations of the of smooth functions, providing a to the manifold.

Tangent spaces and vector fields

In differential geometry, the tangent space at a point p on a smooth manifold M, denoted T_p M, is defined as the set of equivalence classes of smooth curves passing through p, where two curves \gamma and \eta with \gamma(0) = \eta(0) = p are equivalent if their velocities agree upon differentiation in local coordinates, i.e., d(\gamma^i)/dt \big|_{t=0} = d(\eta^i)/dt \big|_{t=0} for all coordinate functions x^i. This construction captures the directions in which the manifold can be instantaneously traversed at p, generalizing the tangent lines to curves in Euclidean space. Equivalently, T_p M can be identified with the space of derivations at p, which are linear maps v: C^\infty(M) \to \mathbb{R} satisfying the Leibniz rule v(fg) = f(p) v(g) + g(p) v(f) for smooth functions f, g; this isomorphism arises because each equivalence class of curves induces a derivation via directional differentiation, and every derivation corresponds to such a class. The tangent bundle TM of M is the \bigcup_{p \in M} T_p M, equipped with a natural \pi: TM \to M sending each to its base point, forming a over M whose fibers are the tangent spaces. Locally, in coordinates (x^1, \dots, x^n) on M, elements of TM are represented as (x, v) with v = v^i \partial/\partial x^i \big|_x, allowing global analysis of directions across the manifold. A X on an U \subset M is a of the restriction of TM to U, assigning to each p \in U a X(p) \in T_p M such that the component functions X^i (in local coordinates) are . Vector fields enable the study of motions and local changes on M, with ensuring compatibility under coordinate transitions. Associated with a vector field X are its integral curves, which are smooth curves \gamma: (-\epsilon, \epsilon) \to M satisfying \gamma'(t) = X(\gamma(t)) for all t in the interval, representing paths tangent to X at every point. Under suitable conditions, such as completeness on compact sets, X generates a local flow \phi_t: U \to M, a one-parameter family of diffeomorphisms satisfying the flow equation \frac{d}{dt} \phi_t(q) = X(\phi_t(q)), \quad \phi_0(q) = q for points q in a neighborhood U, describing the evolution of points along the field's directions over time. The flow provides a global perspective on how X deforms the manifold locally. For two vector fields X and Y, the Lie bracket [X, Y] is defined by [X, Y]f = X(Yf) - Y(Xf) for smooth functions f, yielding another that quantifies the non-commutativity of their flows: if [X, Y] = 0, the flows of X and Y commute, meaning \phi_t \circ \psi_s = \psi_s \circ \phi_t for small t, s. This bracket endows the space of vector fields with a structure, essential for analyzing symmetries and deformations in differential geometry.

Differential forms and exterior derivatives

In differential geometry, a differential k-form on a manifold M is a section of the k-th exterior power of the , \Lambda^k T^*M, which can be viewed as an antisymmetric from the k-fold product of the to the real numbers at each point. These forms generalize scalar functions (0-forms) and covectors (1-forms), providing a coordinate-independent way to describe volumes and orientations. The algebra of differential forms is equipped with the wedge product \wedge, a bilinear operation that combines a p-form \alpha and a q-form \beta into a (p+q)-form \alpha \wedge \beta, satisfying antisymmetry: \alpha \wedge \beta = (-1)^{pq} \beta \wedge \alpha. This product is associative and graded-commutative, forming the exterior algebra \Omega^*(M) = \bigoplus_{k=0}^{\dim M} \Omega^k(M), where \Omega^k(M) denotes the space of smooth k-forms on M. The exterior derivative d: \Omega^k(M) \to \Omega^{k+1}(M) is a linear operator that generalizes the gradient, curl, and divergence in a unified manner, defined locally in coordinates (x^1, \dots, x^n) by d\left( \sum_I f_I \, dx^{i_1} \wedge \cdots \wedge dx^{i_k} \right) = \sum_I df_I \wedge dx^{i_1} \wedge \cdots \wedge dx^{i_k}, where df_I = \sum_j \frac{\partial f_I}{\partial x^j} dx^j and I runs over increasing multi-indices. It satisfies two key properties: nilpotency, d^2 = 0, meaning the exterior derivative of a closed form (one with d\omega = 0) is always zero; and the Leibniz rule, d(\alpha \wedge \beta) = d\alpha \wedge \beta + (-1)^k \alpha \wedge d\beta, for \alpha \in \Omega^k(M). These ensure that exact forms (those of the type d\eta for some \eta) are closed, providing a differential complex. For a smooth map \phi: N \to M between manifolds, the pullback \phi^*: \Omega^k(M) \to \Omega^k(N) induces a homomorphism that preserves the wedge product and commutes with the exterior derivative: \phi^*(\alpha \wedge \beta) = \phi^*\alpha \wedge \phi^*\beta and \phi^*(d\omega) = d(\phi^*\omega). This compatibility allows forms to be transported consistently under manifold maps, facilitating global constructions. The Poincaré lemma states that on a contractible open set U \subset \mathbb{R}^n, every closed k-form is exact, i.e., if d\omega = 0 on U, then there exists \eta \in \Omega^{k-1}(U) such that \omega = d\eta. This local exactness holds more generally on star-shaped domains and underscores the role of topology in distinguishing global from local properties of forms. A central application is de Rham cohomology, which captures the topological invariants of M through the sequence of spaces \Omega^0(M) \xrightarrow{d} \Omega^1(M) \xrightarrow{d} \cdots \xrightarrow{d} \Omega^{\dim M}(M), where the k-th cohomology group is the quotient H^k(M) = \frac{\ker(d|_{\Omega^k(M)})}{\operatorname{im}(d|_{\Omega^{k-1}(M)})}, measuring the failure of closed k-forms to be exact. Introduced by Georges de Rham, these groups are finite-dimensional real vector spaces isomorphic to the singular cohomology groups of M with real coefficients, linking differential geometry to algebraic topology.

Integration on manifolds

Integration on manifolds extends the classical theory of integration from spaces to more general structures, allowing the of integrals of functions and forms over curved spaces. This framework relies on local coordinate charts to define integrals, combined with global tools like partitions of unity to glue local contributions into a manifold-wide integral. An on the manifold is essential, providing a consistent of "positive" direction for integrating top-dimensional forms, which corresponds to a nowhere-vanishing up to sign. For a smooth f on an n-dimensional oriented manifold M, the \int_M f \, dV is defined by covering M with coordinate charts \{U_i, \phi_i\} where \phi_i: U_i \to \mathbb{R}^n, and using a \{\rho_i\} subordinate to this cover such that \sum \rho_i = 1. The is then \int_M f \, dV = \sum_i \int_{\phi_i(U_i)} (\rho_i \circ \phi_i^{-1}) f \circ \phi_i^{-1} \, dx^1 \wedge \cdots \wedge dx^n, where the local is the standard on \mathbb{R}^n. This construction ensures independence from the choice of cover and , as the dictates the sign of the in overlapping charts. More generally, integration applies to k-forms over oriented k-dimensional . For a compact oriented k-submanifold S \subset M and a k-form \omega on M, the \int_S \omega is computed via a parametrization \psi: V \to S from an oriented domain V \subset \mathbb{R}^k, yielding \int_S \omega = \int_V \psi^* \omega, where \psi^* \omega pulls back to a k-form on V integrated against the standard . The in charts accounts for the geometry: if coordinates change from x to y, the transforms as \int_U f \, dx^1 \wedge \cdots \wedge dx^n = \int_{\phi(U)} f \circ \phi^{-1} \left| \det \frac{\partial x}{\partial y} \right| dy^1 \wedge \cdots \wedge dy^n, ensuring consistency across the atlas. A cornerstone result is , which relates the integral of the of a form to the integral: for a compact oriented (n+1)-manifold M with \partial M, and an n-form \omega, \int_M d\omega = \int_{\partial M} \omega. This holds by local computation in charts, where it reduces to the classical on \mathbb{R}^{n+1}, and glues globally via the and orientation compatibility; the d here generalizes the , , and operators. Applications of this integration theory include the Gauss-Bonnet theorem, which equates the integral of the over a compact oriented surface to $2\pi times its , providing a topological via differential geometry; a full treatment appears in the discussion of .

Core Structures

Fiber bundles

In differential geometry, s formalize the attachment of geometric data, such as tangent spaces or frames, to a base manifold in a way that varies smoothly but may exhibit global twisting. The concept originated in with early contributions from Hassler in and was systematically developed by Norman Steenrod, who provided the foundational axiomatic definition. A is a continuous surjective map \pi: E \to M, where E is the total space, M is the base manifold, and F is the typical fiber (a manifold), equipped with an open cover \{U_\alpha\} of M and homeomorphisms (or diffeomorphisms in the smooth case) \phi_\alpha: \pi^{-1}(U_\alpha) \to U_\alpha \times F such that the diagrams commute on overlaps U_\alpha \cap U_\beta, defining transition functions g_{\alpha\beta}: U_\alpha \cap U_\beta \to \mathrm{Diff}(F) (or \mathrm{Homeo}(F)) satisfying the cocycle condition g_{\alpha\beta} \circ g_{\beta\gamma} = g_{\alpha\gamma}. This local triviality ensures that the bundle looks like a product locally, while the transition functions capture the global structure. Principal bundles are a special class where the fiber F is a Lie group G, and G acts freely and transitively on each fiber by right multiplication, making the structure group G itself. Formally, a principal G-bundle is \pi: P \to M with a right G-action on P such that \pi(pg) = \pi(p) for p \in P, g \in G, and the action is free, with local trivializations P|_{U_\alpha} \cong U_\alpha \times G where the transition functions take values in G. These bundles serve as "universal" frames for more general fiber bundles with the same structure group. From a principal G-bundle P \to M and a \rho: G \to \mathrm{GL}(V) of G on a V, one constructs an associated E = P \times_G V \to M, where points are equivalence classes [(p, v)] with (p, v) \sim (pg, \rho(g^{-1})v) for g \in G, and the fiber over each point in M is isomorphic to V. This construction links principal bundles to vector bundles, allowing geometric objects like tangent spaces to be viewed as associated bundles. Prominent examples include the TM \to M of a smooth n-manifold M, a with \mathbb{R}^n where the over p \in M is the T_p M. The LM \to M, or orthonormal if equipped with a , is a principal \mathrm{GL}(n, \mathbb{R})-bundle (or \mathrm{O}(n)-bundle), with the space of bases for \mathbb{R}^n, obtained by associating to points in M. Fiber bundles over a base M with good cover \{U_\alpha\} are classified up to by their clutching functions, which are the functions g_{\alpha\beta}: U_\alpha \cap U_\beta \to G satisfying the cocycle condition; two bundles are isomorphic if their clutching functions differ by a coboundary. For bundles over , this reduces to a single clutching map on the equatorial sphere determining the isomorphism class.

Connections and parallel transport

In differential geometry, on s provide a mechanism for defining of sections and transporting geometric along paths in the base manifold, generalizing the notion of to non-trivial geometries. These structures are essential for studying how fibers over nearby points can be compared systematically. An Ehresmann on a \pi: E \to M with typical fiber F is specified by a subbundle H E \subset T E that is complementary to the vertical subbundle V E = \ker (T \pi), so that T E = H E \oplus V E . This decomposition defines a onto the vertical directions and enables the of tangent vectors at each point in E. The subbundle must be and of equal to the of M. Parallel transport induced by an Ehresmann connection allows the lifting of curves from the base manifold M to horizontal curves in the total space E. For a piecewise smooth curve \gamma: [0,1] \to M with \gamma(0) = p and a point e \in E_p = \pi^{-1}(p), there exists a unique horizontal lift \tilde{\gamma}: [0,1] \to E such that \pi \circ \tilde{\gamma} = \gamma, \tilde{\gamma}(0) = e, and T \tilde{\gamma}(t) \subset H E for all t. The endpoint \tilde{\gamma}(1) \in E_{\gamma(1)} defines the parallel transport map from the fiber E_p to E_{\gamma(1)}, which is a diffeomorphism preserving the fiber structure. This process is path-dependent in general and linear when restricted to vector bundles. The associated with a linear Ehresmann on a E \to M extends to a rule for differentiating . For a \sigma: M \to E and a X on M, the \nabla_X \sigma is a of E defined by (\nabla_X \sigma)(p) = \frac{D}{dt} \big|_{t=0} \sigma(\gamma(t)), where \gamma is an of X at p and the derivative is taken with respect to along \gamma. This operator satisfies Leibniz linearity: \nabla_X (f \sigma) = (X f) \sigma + f \nabla_X \sigma for functions f on M. The full is the map \nabla: \Gamma(T M) \times \Gamma(E) \to \Gamma(E). Holonomy groups quantify the obstruction to path-independence in parallel transport, arising from the geometry of the bundle and connection. The holonomy group at a base point p \in M is the subgroup of diffeomorphisms of the fiber E_p generated by parallel transport maps along all loops based at p; the restricted holonomy group considers only loops contractible in M. For linear connections on vector bundles, the holonomy group embeds into the general linear group \mathrm{GL}(F), where F is the typical fiber, and measures the cumulative effect of transporting vectors around closed paths. Charles Ehresmann introduced these concepts in the 1950s to generalize Cartan's affine connections to arbitrary fiber bundles. The of a captures the failure of to commute along non-commuting vector fields and is expressed via the curvature form. For vector fields X, Y on M and a \sigma of the , the operator is given by \Omega(X,Y) \sigma = [\nabla_X, \nabla_Y] \sigma - \nabla_{[X,Y]} \sigma, where [\nabla_X, \nabla_Y] denotes the of the operators. This \Omega is a tensorial map \Omega \in \Gamma(T^*M \otimes T^*M \otimes \mathrm{End}(E)), alternating in X and Y, and vanishes if and only if the is flat, meaning depends only on classes of paths. In the context of Ehresmann connections, the curvature form measures the integrability of the horizontal distribution.

Riemannian metrics

In differential geometry, a Riemannian metric provides a way to measure lengths, angles, and distances intrinsically on a smooth manifold, generalizing the structure to curved spaces. Introduced by in his foundational 1854 lecture, the metric allows geometry to vary smoothly from point to point, enabling the study of non-Euclidean spaces without reference to an . Formally, a Riemannian metric g on a smooth manifold M is a smooth section of the bundle \mathrm{Sym}^2(T^*M) of symmetric bilinear forms on the tangent bundle TM, such that g_p is positive definite on each tangent space T_pM for p \in M. This means g assigns to every point p an inner product g_p: T_pM \times T_pM \to \mathbb{R} that varies smoothly, satisfying g_p(v, v) > 0 for all nonzero v \in T_pM and g_p(u, v) = g_p(v, u). In local coordinates (x^1, \dots, x^n) around p, g takes the form g = \sum_{i,j=1}^n g_{ij}(x) \, dx^i \otimes dx^j, where (g_{ij}(x)) is a symmetric positive definite matrix at each point. The Riemannian induces a notion of for piecewise smooth curves \gamma: [a, b] \to M. The L(\gamma) is defined as L(\gamma) = \int_a^b \sqrt{g_{\gamma(t)}(\gamma'(t), \gamma'(t))} \, dt, which measures the infinitesimal along the curve using the local inner product. This integral is independent of parametrization up to reparametrization and allows minimization problems, such as finding shortest paths known as . Associated to any Riemannian g is the \nabla, a unique on TM that is both torsion-free (\nabla_X Y - \nabla_Y X = [X, Y] for vector fields X, Y) and metric-compatible (X(g(Y, Z)) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z) for all vector fields X, Y, Z). This connection, introduced by in 1917, enables of vectors along curves while preserving the metric structure. are the "straight lines" in this geometry, defined as curves \gamma satisfying \nabla_{\gamma'} \gamma' = 0. In local coordinates, this yields the geodesic equation \frac{d^2 x^k}{dt^2} + \Gamma^k_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt} = 0, where \Gamma^k_{ij} are the determined by g via \Gamma^k_{ij} = \frac{1}{2} g^{kl} (\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}). Classic examples illustrate these concepts. On \mathbb{R}^n, the standard Riemannian is g = \sum_{i=1}^n (dx^i)^2, yielding straight-line distances and zero , so geodesics are linear. On the 2-sphere S^2 embedded in \mathbb{R}^3, the round induced from the ambient is g = d\theta^2 + \sin^2\theta \, d\phi^2 in spherical coordinates, with geodesics as great circles of constant speed. Similarly, the hyperbolic plane admits a like the Poincaré upper half-plane model g = \frac{dx^2 + dy^2}{y^2} for y > 0, where geodesics are semicircles orthogonal to the boundary or vertical lines, demonstrating constant negative in a non- setting.

Curvature tensors

In Riemannian geometry, curvature tensors quantify the intrinsic deviation of a manifold from being flat, capturing how parallel transport around closed loops fails to return vectors unchanged. The Riemann curvature tensor serves as the fundamental object encoding this information, arising naturally from the Levi-Civita connection associated with the Riemannian metric. The Riemann curvature tensor R on a Riemannian manifold (M, g) with Levi-Civita connection \nabla is defined for vector fields X, Y, Z by R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z. This operator measures the non-commutativity of second covariant derivatives, adjusted for the Lie bracket of the vector fields; it vanishes if and only if the manifold is locally Euclidean. The tensorial nature of R ensures it is independent of local coordinates, and its components satisfy symmetries such as R(X,Y) = -R(Y,X) and R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0. A key invariant derived from the Riemann tensor is the sectional curvature, which describes the curvature of 2-dimensional subspaces. For an orthonormal pair of vectors X, Y spanning a plane \sigma in the tangent space, the sectional curvature is K(\sigma) = \langle R(X,Y)Y, X \rangle. More generally, for linearly independent X, Y, K(\sigma) = \frac{\langle R(X,Y)Y, X \rangle}{|X|^2 |Y|^2 - \langle X, Y \rangle^2}. Sectional curvatures determine the full Riemann tensor via polarization, and constant sectional curvature classifies spaces of constant curvature, such as spheres (K > 0) or hyperbolic spaces (K < 0). Riemann introduced this concept in his foundational 1868 work on manifold geometry. Contractions of the Riemann tensor yield the Ricci tensor and scalar curvature, which average the sectional curvatures. The Ricci tensor \mathrm{Ric} is defined by \mathrm{Ric}(X,Y) = \sum_i \langle R(E_i, X)Y, E_i \rangle, where \{E_i\} is an orthonormal basis of the tangent space; it is symmetric and thus diagonalizable. The scalar curvature \mathrm{scal} is the trace of the Ricci tensor, \mathrm{scal} = \sum_i \mathrm{Ric}(E_i, E_i). These quantities encode volume distortion under geodesic flow and play central roles in variational problems like the Einstein-Hilbert action. The Ricci tensor was developed by Gregorio Ricci-Curbastro in his absolute differential calculus. The Bianchi identities impose essential constraints on the curvature tensors. The first Bianchi identity, algebraic in nature, R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0, follows from the torsion-freeness of the Levi-Civita connection and holds for any affine connection without torsion. The second Bianchi identity, \nabla_W R(X,Y)Z + \nabla_X R(Y,W)Z + \nabla_Y R(W,X)Z = 0, is differential and implies conservation laws, such as the contracted form \nabla^* \mathrm{Ric} = \frac{1}{2} \nabla \mathrm{scal}, linking geometry to analysis on manifolds. These identities were first derived by Ricci in the late 19th century. For compact oriented surfaces without boundary, the Gauss-Bonnet theorem relates total scalar curvature to topology: \int_M \mathrm{scal} \, d\mathrm{vol}_g = 2\pi \chi(M), where \chi(M) is the Euler characteristic and d\mathrm{vol}_g is the volume form induced by the metric. With boundary, geodesic curvature and angle terms appear to maintain the equality. This theorem, bridging local geometry and global invariants, was proved by Bonnet in 1848 for geodesic polygons, building on Gauss's earlier work from 1827.

Branches

Pseudo-Riemannian geometry

Pseudo-Riemannian geometry generalizes the framework of by allowing metrics that are indefinite, enabling the mathematical description of spacetimes with both spatial and temporal dimensions in physical theories like . A pseudo-Riemannian manifold consists of a smooth manifold M of dimension n together with a smooth metric tensor g, which is a non-degenerate symmetric bilinear form on the tangent spaces T_p M at each point p \in M. Unlike the positive-definite metrics in , the pseudo-Riemannian metric has a signature (p, q) where p + q = n, p is the number of positive eigenvalues, and q is the number of negative eigenvalues of the metric in local coordinates. This signature determines the type of geometry; for instance, the Lorentzian signature (1, n-1) is prevalent in relativistic contexts, with one timelike direction and n-1 spacelike directions. The indefinite nature of the metric induces a causal structure on the manifold, classifying tangent vectors v \in T_p M based on the sign of the quadratic form g(v, v): a vector is timelike if g(v, v) < 0, spacelike if g(v, v) > 0, and null (or lightlike) if g(v, v) = 0. At each point p, the set of null vectors forms the light cone, a double cone consisting of future and past lightlike directions that separates the timelike interior from the spacelike exterior. This structure is crucial for defining in physical models, as timelike curves represent possible worldlines of massive particles, null curves model light rays, and spacelike curves connect simultaneous events. In Lorentzian manifolds, a consistent choice of future-directed timelike vectors allows for a time , ensuring global consistency in causal relations. Geodesics on pseudo-Riemannian manifolds are defined via the associated to g, satisfying the geodesic equation \frac{D}{dt} \dot{\gamma} = 0 for a \gamma. Due to the indefinite , the arc-length parameter \int \sqrt{|g(\dot{\gamma}, \dot{\gamma})|} \, dt may be imaginary or zero, so an affine t is used instead, normalizing the equation without assuming . Timelike geodesics parametrized affinely correspond to for observers, null geodesics to light propagation, and spacelike geodesics to extremal spatial paths; however, maximality or minimality of lengths is not guaranteed as in the Riemannian case, leading to potential conjugate points along timelike or null rays. The curvature in pseudo-Riemannian geometry is captured by the Riemann curvature tensor R, constructed identically to the Riemannian setting from the connection, measuring the failure of parallel transport around loops. While the algebraic properties and Bianchi identities remain the same, the causal structure influences interpretations: for example, sectional curvatures along timelike or null planes determine geodesic deviation and focusing behavior in spacetimes, affecting phenomena like gravitational lensing or singularity formation. This adaptation highlights how indefinite metrics introduce hyperbolic-like behaviors absent in positive-definite cases. A primary application lies in , where 4-dimensional manifolds model , governed by the : G_{\mu\nu} = 8\pi T_{\mu\nu}, with the G_{\mu\nu} derived from the Ricci tensor and of g, equating curvature to the stress-energy-momentum tensor T_{\mu\nu} of matter and fields (in units where c = G = 1). These equations, formulated on pseudo-Riemannian manifolds, encapsulate the equivalence of and .

Finsler geometry

Finsler geometry generalizes by replacing the inner product on with a more general that can depend on the direction of vectors, allowing for anisotropic structures. A Finsler metric on a smooth manifold M is a function F: TM \to [0, \infty) that is smooth and positive away from the zero section, positively homogeneous of degree 1 in the fiber variables (i.e., F(x, \lambda y) = \lambda F(x, y) for \lambda > 0 and y \in T_x M), and strongly convex, meaning that for each x \in M and y \in T_x M \setminus \{0\}, the Hessian matrix g_{ij}(x,y) = \frac{1}{2} \frac{\partial^2 F^2(x,y)}{\partial y^i \partial y^j} defines a positive definite quadratic form on T_x M. This convexity ensures that F behaves like a on each , but unlike Riemannian metrics, F need not derive from an inner product and can be asymmetric. A Finsler manifold is a pair (M, F) where M is a smooth manifold and F is a Finsler metric on it. Riemannian manifolds form a special case where F(x,y) = \sqrt{g_{ij}(x) y^i y^j} for a Riemannian g_{ij}(x), independent of direction beyond the . In Finsler geometry, the fundamental structure is provided by a spray, a homogeneous S on TM of degree 1 that governs flow. The spray is expressed as S = y^i \frac{\partial}{\partial x^i} - 2 G^i(x,y) \frac{\partial}{\partial y^i}, where the spray coefficients G^i satisfy G^i(x, \lambda y) = \lambda^2 G^i(x,y) and are determined by the metric via G^i = \frac{1}{2} g^{ij} (y^k \frac{\partial g_{jk}}{\partial x^l} y^l - \frac{\partial g_{jk}}{\partial y^l} y^l y^k), or more directly from the metric's Hilbert form. Finsler connections, such as the Chern connection or the Berwald connection, are defined to be compatible with both the metric and the volume form induced by \det(g_{ij}), but they generally differ from the due to the directional dependence. Geodesics on a Finsler manifold are curves \gamma: I \to M that locally minimize the arc length \int_I F(\gamma(t), \dot{\gamma}(t)) \, dt, arising from Hilbert's variational problem in the calculus of variations for non-quadratic Lagrangians. The geodesic equations are second-order ODEs projected from the spray's integral curves: \frac{d^2 \gamma^i}{dt^2} + 2 G^i(\gamma, \dot{\gamma}) = 0. Curvature in Finsler geometry is measured by the flag curvature K^P(x, y, V), which generalizes the sectional curvature of Riemannian geometry; for a flag P spanned by a plane containing the direction y \in T_x M \setminus \{0\} and a transverse vector V \in T_x M, it is defined as K^P(x, y, V) = \frac{g_y(V, RY(V))}{g_y(V,V) g_y(y,y) - [g_y(y,V)]^2}, where R is the curvature operator of the connection, and g_y is the metric tensor at y. Positive flag curvature implies comparison properties analogous to those in spaces of constant sectional curvature. Prominent examples include Randers metrics, of the form F(x,y) = \alpha(x,y) + \beta(x,y), where \alpha = \sqrt{g_{ij} y^i y^j} is a Riemannian metric and \beta = b_i y^i is a 1-form with \|\beta\|_x < 1, which arise in the Zermelo navigation problem modeling motion in a medium with wind or current. These metrics are projectively flat under certain conditions and have been studied for their curvature properties since their introduction by Gunnar Randers in 1941.

Symplectic geometry

Symplectic geometry is a branch of that studies manifolds equipped with a symplectic structure, which provides a geometric framework for classical mechanics, particularly Hamiltonian systems. A symplectic manifold is defined as a pair (M, \omega), where M is a smooth even-dimensional manifold and \omega \in \Omega^2(M) is a closed non-degenerate 2-form, meaning d\omega = 0 and the top power \omega^n \neq 0 at every point, where $2n = \dim M. This non-degeneracy ensures that \omega induces a volume form \omega^n and that the pairing \omega_x: T_x M \times T_x M \to \mathbb{R} is non-degenerate, making (T_x M, \omega_x) a symplectic vector space. The closedness condition d\omega = 0 reflects the conservation of the symplectic structure under Hamiltonian flows, distinguishing it from more general almost symplectic forms. A fundamental local normal form for symplectic manifolds is given by the Darboux theorem, which asserts that for any point p \in M, there exist local coordinates (q_1, \dots, q_n, p_1, \dots, p_n) around p such that \omega = \sum_{i=1}^n dq_i \wedge dp_i. These coordinates, often called canonical or Darboux coordinates, model the phase space of Hamiltonian mechanics, where q_i represent generalized positions and p_i generalized momenta, and the symplectic form captures the canonical commutation relations. This theorem implies that symplectic manifolds have no local invariants beyond their dimension, in stark contrast to Riemannian geometry where curvature provides such invariants. Central to symplectic geometry is the notion of Hamiltonian vector fields, which arise from smooth functions H: M \to \mathbb{R} serving as Hamiltonians. The Hamiltonian vector field X_H is uniquely determined by the equation \iota_{X_H} \omega = -dH, where \iota denotes the interior product. This defines a Lie algebra homomorphism from the space of functions to vector fields, with the flow of X_H preserving \omega and generating canonical transformations in mechanics. The Poisson bracket on functions is then defined by \{f, g\} = \omega(X_f, X_g), which equals X_f g = -X_g f and measures the directional derivative along Hamiltonian flows. This bracket endows the space of smooth functions C^\infty(M) with a Lie algebra structure, satisfying bilinearity, antisymmetry \{f, g\} = -\{g, f\}, the Leibniz rule \{f, gh\} = g\{f, h\} + h\{f, g\}, and the Jacobi identity \{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0, the latter following from the closedness of \omega via Cartan's magic formula. On compact manifolds, the Moser theorem provides a stability result for symplectic structures, stating that if (\omega_t)_{t \in [0,1]} is a smooth family of symplectic forms with [\omega_t] = [\omega_0] in the de Rham cohomology H^2(M; \mathbb{R}), then there exists a smooth family of diffeomorphisms (\phi_t)_{t \in [0,1]} with \phi_0 = \mathrm{id} such that \phi_t^* \omega_t = \omega_0 for all t. This isotopy theorem implies that symplectic forms in the same cohomology class are symplectomorphic, allowing deformation without altering the underlying topology, and it underpins many rigidity and classification results in symplectic topology. The proof relies on solving a time-dependent homotopy equation for vector fields generating the isotopy, leveraging the compactness to ensure solvability.

Contact geometry

Contact geometry is a branch of differential geometry that examines structures on odd-dimensional smooth manifolds, particularly hyperplane distributions in the tangent bundle endowed with a conformal symplectic property. These structures, known as contact structures, arise as the kernels of contact 1-forms and provide an odd-dimensional analogue to symplectic geometry, capturing non-integrability in a maximal sense. This framework is pertinent to various physical contexts, including the modeling of thermodynamic phase spaces and the geometry of optical rays. A contact structure \xi on a smooth manifold M of dimension $2n+1 is defined as the kernel of a 1-form \alpha \in \Omega^1(M), termed a contact form, satisfying the non-degeneracy condition that d\alpha restricts to a symplectic form on \xi = \ker \alpha. Specifically, \alpha \wedge (d\alpha)^n \neq 0 everywhere on M, ensuring that \xi has maximal rank $2nand is nowhere integrable. Two contact forms\alphaand\betadefine the same structure if\beta = f \alphafor some nowhere-vanishing smooth functionf: M \to \mathbb{R}^\times$. Contactomorphisms are diffeomorphisms \phi: (M, \xi) \to (M, \xi) that preserve the contact structure, meaning \phi_* \xi_p = \xi_{\phi(p)} for all p \in M. In terms of contact forms, this equates to \phi^* \alpha = \lambda \alpha for some positive function \lambda: M \to \mathbb{R}^+, reflecting the conformal invariance of the structure. The group of contactomorphisms acts on the space of contact structures, facilitating local and global classifications. To each contact form \alpha is associated the Reeb vector field R_\alpha, a unique smooth vector field on M satisfying the conditions \alpha(R_\alpha) = 1 and \iota_{R_\alpha} d\alpha = 0. This field is transverse to the contact distribution \xi and generates a flow that preserves \alpha up to scaling, playing a central role in the dynamics and topology of contact manifolds. The Reeb flow encodes the "symplectic" direction orthogonal to \xi. The Darboux theorem for contact manifolds asserts that every contact structure is locally standard: around any point p \in M, there exist Darboux coordinates (x_1, \dots, x_n, y_1, \dots, y_n, z) such that \alpha = dz - \sum_{i=1}^n y_i \, dx_i. This normal form highlights the universality of local contact geometry and parallels the in . Legendrian submanifolds are integral submanifolds of the contact distribution \xi of maximal dimension n, meaning they are tangent to \xi and isotropic with respect to the conformal symplectic form induced by d\alpha|_{\xi}. These submanifolds, often arising as fronts or caustics in applications, are key objects in , with their knots and links exhibiting invariants distinct from classical .

Complex and Kähler geometry

Complex geometry emerges as a branch of differential geometry that endows smooth manifolds with a compatible complex structure, enabling the application of tools from complex analysis. A complex manifold is defined as a smooth manifold M equipped with an almost complex structure J: TM \to TM, a smooth bundle endomorphism satisfying J^2 = -\mathrm{Id}, which is integrable in the sense that its Nijenhuis tensor vanishes. Integrability ensures the existence of local holomorphic coordinates where J acts as multiplication by i on the tangent spaces, as established by the Newlander-Nirenberg theorem. This structure partitions the cotangent bundle into holomorphic and anti-holomorphic parts, facilitating the study of holomorphic functions and vector bundles on M. A key advancement integrates with complex manifolds through Hermitian metrics. A Hermitian metric g on a complex manifold (M, J) is a satisfying g(JX, JY) = g(X, Y) for all vector fields X, Y, making it compatible with the complex structure. The associated fundamental (or Kähler) form is the closed symplectic 2-form \omega(X, Y) = g(JX, Y). If \omega is closed (d\omega = 0), then g defines a , and (M, g, J, \omega) forms a . This closure condition implies that \omega is locally expressible via a \phi, a real-valued function such that \omega = i \partial \bar{\partial} \phi in holomorphic coordinates, where \partial and \bar{\partial} are the . The geometry of Kähler manifolds is enriched by curvature considerations, particularly the Ricci form, which captures essential topological information. For a Kähler metric g with local expression g = g_{j\bar{k}} dz^j \otimes d\bar{z}^k, the Ricci form is \rho = -i \partial \bar{\partial} \log \det(g_{j\bar{k}}), representing the first Chern class c_1(M) up to a factor of $2\pi. This form plays a pivotal role in prescribing metrics with desired curvature properties. The Calabi-Yau theorem asserts that on a compact Kähler manifold with vanishing first Chern class (c_1(M) = 0), there exists a unique Kähler metric in each Kähler class with prescribed Ricci form, including Ricci-flat metrics when the target Ricci form is zero. This existence result, resolving a conjecture by Eugenio Calabi, has profound implications for Ricci-flat Kähler metrics, known as Calabi-Yau metrics. A canonical example of a Kähler manifold is complex projective space \mathbb{CP}^n, equipped with the Fubini-Study metric, which arises as the quotient of the unit sphere in \mathbb{C}^{n+1} by the U(1)-action. This metric is Kähler-Einstein, with constant positive holomorphic sectional curvature, and its Kähler form integrates to \pi over \mathbb{CP}^1. The Fubini-Study metric provides a model for homogeneous Kähler geometry and underlies the study of projective varieties.

Conformal geometry

Conformal geometry is a branch of differential geometry that investigates structures on manifolds invariant under conformal transformations, which are diffeomorphisms preserving angles between curves but allowing lengths to scale by a positive factor. These transformations act multiplicatively on the tangent spaces, preserving the causal structure defined by light cones in pseudo-Riemannian settings but focusing primarily on Riemannian cases. The core object is a conformal structure, defined as an equivalence class of Riemannian metrics on a smooth manifold M, where two metrics g and g' belong to the same class if there exists a smooth positive function f such that g' = e^{2f} g. This scaling relation ensures that angles, measured via the inner product up to conformal factor, remain unchanged, while the class encodes the geometry modulo local rescalings. To develop differential geometry within a conformal class, Weyl structures provide a framework for connections that respect the conformal invariance. A Weyl structure consists of a choice of representative metric g from together with a 1-form ω, defining the Weyl connection ∇^W by ∇^W_X Y = ∇^g_X Y - ω(X) Y - ω(Y) X, where ∇^g is the Levi-Civita connection of g; this connection is torsion-free and satisfies the modified metricity condition ∇^W g = -2 ω ⊗ g. Unlike the Levi-Civita connection, parallel transport along curves with respect to ∇^W scales lengths by factors depending on the integral of ω, reflecting the non-integrable nature of the scale connection. The curvature of ∇^W decomposes into the Riemannian curvature plus terms involving the scale curvature, a tensor F = dω + ω ∧ ω that measures the obstruction to integrating the scale structure locally; in dimensions greater than 2, the Weyl tensor, the trace-free part of the curvature, is a key conformal invariant. Higher-order conformal invariants arise through differential operators that transform covariantly under rescalings. The Paneitz operator, a fourth-order operator on functions in dimension 4, exemplifies this: for a metric g, it acts as P_g φ = Δ_g^2 φ + div_g (W_g ∇g φ) - (1/2) Scal_g Δ_g φ + 2 |W_g|^2 φ, where Δ_g is the , W_g the , and Scal_g the ; under g' = e^{2f} g, it transforms as P{g'} (e^{-4f} ψ) = e^{4f} P_g ψ, ensuring bi-degree (0,4) covariance. Introduced in an unpublished 1983 manuscript by S. Paneitz and independently rediscovered, it generalizes the second-order and plays a central role in prescribing Q-curvature, a conformally invariant integral of a fourth-order Paneitz curvature density. In higher even dimensions, analogous extend this construction. For asymptotically hyperbolic manifolds, relevant in general relativity and holography, the Fefferman-Graham expansion provides a formal series solution to the Einstein equations near the conformal boundary. Consider a manifold-with-boundary (M, ∂M) with a defining function ρ such that ρ = 0 on ∂M and dρ ≠ 0 there; an asymptotically hyperbolic metric g satisfies |g + ρ^2 h| = O(ρ^3) near ∂M for some metric h on ∂M, making [g/ρ^2]|_{∂M} a conformal class on the boundary. The expansion writes g = ρ^2 \bar{g}_0 + ρ^4 \bar{g}2 + \cdots + ρ^n \bar{g}{n-1} + O(ρ^{n+1}), where even powers are determined by the boundary data \bar{g}_0 via obstructions, and odd powers above the first relate to stress-energy tensors in holographic contexts; this was developed in foundational work on conformally compact Einstein metrics. Tractor bundles offer a powerful tool for constructing and computing conformal invariants systematically. The standard tractor bundle \mathcal{A}M over a conformal manifold (M,) of dimension n is a rank (n+2) vector bundle with transition functions induced by the projective representation of the conformal group CO(n+1,1); sections transform covariantly, with a distinguished subbundle isomorphic to the weighted tangent bundle. Introduced by T. Thomas in the 1930s as structure bundles for conformal invariants and reformulated in modern terms, the bundle admits a canonical tractor connection ∇^T compatible with a Lorentzian tractor metric h of signature (n+1,1), enabling the definition of conformally invariant powers of differential operators like the Dirac operator or powers of the Laplacian via tractor calculus. Applications include deriving conserved currents, embedding conformal structures into higher-dimensional ambient spaces, and studying boundary invariants in Fefferman-Graham settings.

Geometric analysis

Geometric analysis employs analytic techniques, particularly partial differential equations (PDEs) and geometric flows, to investigate the properties and evolution of and submanifolds, focusing on aspects such as curvature evolution, singularities, and optimization problems. This approach bridges differential geometry with analysis, leveraging tools like variational methods and heat-type equations to derive qualitative and quantitative results about geometric structures. Central to the field is the study of flows that deform geometric objects in a way that decreases energy functionals, such as area or volume, while preserving certain invariants. One foundational tool is the mean curvature flow, which evolves a hypersurface F: M \times [0, T) \to \mathbb{R}^{n+1} according to the equation \frac{\partial F}{\partial t} = H \nu, where H is the mean curvature scalar and \nu is the unit outward normal vector. Introduced by Gerhard Huisken in his seminal 1984 work, this flow minimizes the area of the hypersurface by moving it in the direction of its mean curvature, analogous to the heat equation smoothing out irregularities. For convex hypersurfaces in Euclidean space, the flow contracts the surface to a point in finite time, with asymptotic behavior approaching a round sphere scaled by the remaining volume. This evolution reveals singularities, such as neckpinch phenomena, where the surface develops high curvature regions before collapsing. Similarly, the Ricci flow, developed by Richard Hamilton in 1982, deforms the metric tensor g on a Riemannian manifold (M, g) via the PDE \frac{\partial g}{\partial t} = -2 \operatorname{Ric}(g), where \operatorname{Ric}(g) is the Ricci curvature tensor. This parabolic equation evolves the geometry to make curvature more uniform, with the goal of simplifying the manifold's structure; for example, on spheres, it preserves the round metric up to scaling. To handle non-compactness or volume expansion, a normalized version \frac{\partial g}{\partial t} = -2 \operatorname{Ric}(g) + \frac{r}{n} g is often used, where r is the average scalar curvature and n = \dim M. Hamilton's framework laid the groundwork for resolving the through singularity analysis, though the flow can develop singularities in finite time, necessitating surgical modifications to continue the evolution. The Yamabe problem seeks a conformal metric \tilde{g} = u^{4/(n-2)} g on an n-dimensional manifold with constant scalar curvature, reducing to solving the PDE \Delta_g u - \frac{n-2}{4(n-1)} R_g u + \frac{n-2}{4(n-1)} \tilde{R} u^{(n+2)/(n-2)} = 0, where R_g and \tilde{R} are the scalar curvatures. Posed by Hidehiko Yamabe in 1960, the problem was affirmatively solved in the 1980s through contributions by Neil Trudinger (for subcritical cases), Thierry Aubin (using sub- and super-solutions for positive Yamabe invariant), and Richard Schoen (via positive mass theorem for the critical case on spheres). This resolution highlights the existence of constant scalar curvature metrics in every conformal class, with applications to understanding the topology via curvature obstructions. Minimal surfaces, which locally minimize area and have zero mean curvature, arise in the Plateau problem: given a Jordan curve in \mathbb{R}^3, find a surface of least area spanning it. Solved independently by Jesse Douglas and Tibor Radó in 1931 using Perron's method and Dirichlet's principle, the existence is guaranteed for smooth boundaries, with the solution being a disk-type surface under suitable conditions. A key rigidity result is the Bernstein theorem, proved by Sergei Bernstein in 1915–1917, stating that any entire minimal graph over \mathbb{R}^2 in \mathbb{R}^3 is a plane; this extends to higher dimensions up to n=7 by Charles Morrey and James Bombieri–Enrico De Giorgi–Enrico Giusti, but fails in dimension 8 due to counterexamples by Eberhard Hopf and William Meeks–Willy Müller. These results underscore the flatness of low-dimensional minimal hypersurfaces. Post-2000 advances in singularity models for these flows have refined our understanding of geometric evolution beyond smooth regimes. In Ricci flow, Grigori Perelman's 2002 entropy functional and monotonicity formula enabled the classification of ancient solutions and singularity profiles, culminating in the proof of the via Ricci flow with surgery. For mean curvature flow, works by Simon Brendle and Karsten Grove (2009) and Tobias Colding and William Minicozzi (2010s) identified shrinker and expander models as tangent limits at singularities, with quantitative non-collapsing estimates ensuring controlled behavior; recent results, such as those by Bruce Kleiner and John Lott (2020s), further delineate type I and type II singularities in higher codimensions. These models provide asymptotic descriptions, linking local blow-up dynamics to global topology.

Gauge theory

Gauge theory provides a geometric framework within differential geometry for modeling the fundamental forces of particle physics, particularly through the use of principal bundles and their connections. In this context, a gauge theory is formulated on a principal bundle P \to M with structure group G, a compact Lie group, where the base manifold M represents spacetime. The gauge group G encodes the internal symmetries of the theory, such as the unitary group SU(3) for quantum chromodynamics or SU(2) \times U(1) for electroweak interactions. The connection A, known as the gauge potential, is a Lie algebra-valued 1-form on P with values in the Lie algebra \mathfrak{g} of G, which locally describes the parallel transport of fibers and locally resembles a matrix-valued potential in physics. The curvature F of the connection A, which measures the failure of parallel transport around closed loops, is given by the 2-form F = dA + A \wedge A, where the wedge product incorporates the Lie bracket in \mathfrak{g}. This expression generalizes the electromagnetic field strength tensor and satisfies the Bianchi identity d_A F = 0, with d_A = d + [A, \cdot] denoting the covariant exterior derivative. The dynamics of the gauge field are governed by the Yang-Mills action functional, S = \int_M \operatorname{tr}(F \wedge *F), where \operatorname{tr} is a G-invariant inner product on \mathfrak{g} (often negative the Killing form), and * is the Hodge star operator induced by a metric on M. This action, introduced by Yang and Mills, leads to the Euler-Lagrange equations d_A *F = 0, which are nonlinear partial differential equations modeling force-carrying bosons. Special solutions known as instantons arise as self-dual or anti-self-dual connections, satisfying F = \pm *F, which minimize the Yang-Mills action on compactified Euclidean spacetime \mathbb{R}^4 \cup \{\infty\} \cong S^4. These finite-action solutions, first constructed explicitly for SU(2), have topological charge given by the second Chern number \int_M \operatorname{tr}(F \wedge F)/(8\pi^2), an integer classifying the bundle. The Atiyah-Singer index theorem computes the dimension of the moduli space of instantons via the index of the Dirac operator coupled to A, relating analytic properties to topological invariants like the Pontryagin class.91351-2) Magnetic monopoles emerge in gauge theories with spontaneous symmetry breaking, realized as 't Hooft-Polyakov solitons in the Georgi-Glashow model with gauge group SU(2) broken to U(1) by a Higgs field in the adjoint representation. These finite-energy configurations carry magnetic charge and asymptotically resemble Dirac monopoles, with the Higgs field providing a smooth core to avoid singularities. In seven-dimensional gauge theories on manifolds with G_2 holonomy, BPS monopoles satisfy Bogomol'nyi-Prasad-Sommerfield (BPS) equations, F = \varphi \wedge \varphi coupled to a triplet Higgs \varphi, saturating a topological bound on the energy and preserving half the supersymmetry in string theory compactifications.90453-1)00217-1) More recently, Seiberg-Witten invariants have revolutionized the study of smooth 4-manifolds by defining gauge-theoretic invariants via solutions to the Seiberg-Witten equations, which perturb the Yang-Mills-Dirac system: d_A \phi = 0 for a spinor \phi and F^+ = \sigma(\phi) for the self-dual part of the curvature, where \sigma is a quadratic map. These invariants, counting zeros of the map from connections to spinors modulo gauge, detect smooth structures and symplectic forms, with applications to Donaldson invariants via blow-up formulas.

Approaches

Intrinsic geometry

In differential geometry, intrinsic geometry examines the properties of a Riemannian manifold that depend solely on its metric tensor, enabling the definition of distances, angles, and curvatures through internal measurements without reliance on any ambient space. The Riemannian metric g, a smoothly varying inner product on the tangent spaces, serves as the foundational structure, allowing coordinate-free computations of lengths via the norm \|v\|_g = \sqrt{g(v,v)} and angles via the cosine formula \cos \theta = \frac{g(u,v)}{\|u\|_g \|v\|_g}. Distances are then realized as infima of lengths of curves connecting points, leading to the geodesic distance d(p,q) = \inf \{ L(\gamma) \mid \gamma(0)=p, \gamma(1)=q \}, where L(\gamma) = \int_0^1 \sqrt{g(\dot{\gamma},\dot{\gamma})} \, dt. This framework ensures all geometric invariants are preserved under isometries of the metric. A cornerstone of intrinsic geometry is the Theorema Egregium, established by Carl Friedrich Gauss, which asserts that the Gaussian curvature K of a surface is intrinsically determined by the first fundamental form I = g_{ij} \, dx^i dx^j, the metric tensor in local coordinates. Specifically, K can be expressed as a quotient of determinants involving partial derivatives of the metric coefficients, such as K = \frac{1}{EG - F^2} \left( \frac{\partial \Gamma_{12}^1}{\partial u} - \frac{\partial \Gamma_{11}^2}{\partial v} + \cdots \right), where E, F, G are components of I and \Gamma are Christoffel symbols derived from g; this shows K remains invariant under local isometries that preserve I. Gauss's result revolutionized the field by proving that certain extrinsic notions, like bending without stretching, do not alter intrinsic curvature, laying the groundwork for abstract manifold theory. The intrinsic Laplacian operator, central to analysis on manifolds, is defined as \Delta f = \mathrm{div}(\mathrm{grad} f), where \mathrm{grad} f is the metric gradient satisfying g(\mathrm{grad} f, X) = df(X) for vector fields X, and divergence uses the volume form induced by g. This operator appears in the heat equation \frac{\partial u}{\partial t} = \Delta u, which models diffusion intrinsically on the manifold, with solutions given by the heat kernel p_t(x,y) that satisfies (\partial_t - \Delta_x) p_t(x,y) = 0 and integrates to 1 over the manifold. The equation's properties, such as uniqueness and smoothness of solutions, hold on complete Riemannian manifolds and underpin spectral geometry, including eigenvalue estimates for -\Delta. Comparison theorems further illuminate intrinsic geometry by relating manifold behavior to model spaces via curvature bounds. The Rauch comparison theorem, for instance, compares Jacobi fields along : if a manifold M^n has sectional s K_M \leq K_N compared to another manifold N^n, then for unit-speed \gamma, \tilde{\gamma} with corresponding Jacobi fields J, \tilde{J} satisfying matching initial conditions (e.g., J(0)=0, \nabla_{\dot{\gamma}} J(0) = V), the inequality |J(t)| \geq |\tilde{J}(t)| holds for small t > 0, with equality in constant cases. This implies bounds on conjugate points and growth, such as faster geodesic spreading in negative relative to . Élie Cartan extended intrinsic geometry beyond Riemannian metrics to Cartan geometries, which model manifolds on homogeneous spaces G/H via a principal H-bundle equipped with a Cartan connection—a \mathfrak{g}-valued 1-form generalizing the Levi-Civita connection by incorporating the full Lie algebra \mathfrak{g} of a Lie group G. The curvature form \Omega = d\omega + \frac{1}{2} [\omega, \omega], where \omega is the connection, measures deviation from flatness intrinsically, unifying structures like conformal and projective geometries under a Klein geometry framework. Cartan's approach, developed in the early 20th century, emphasizes moving frames and equivalence methods to classify such geometries.

Extrinsic geometry

Extrinsic geometry examines submanifolds in a higher-dimensional , focusing on how the influences the geometry through the interaction between the and the normal bundle. Unlike intrinsic approaches that rely solely on the , extrinsic geometry incorporates the ambient space's flat structure to define quantities like the second fundamental form, which captures the "bending" of the . This perspective is essential for understanding objects, such as surfaces in \mathbb{R}^3, where normal directions provide additional information about . The second fundamental form, introduced by Gauss in his 1827 work on curved surfaces, measures the extrinsic curvature by projecting the covariant derivative onto the normal space. For a submanifold M \subset \mathbb{R}^n with induced Levi-Civita connection \nabla, and tangent vectors X, Y \in T_pM, it is defined as \mathrm{II}(X, Y) = (\nabla_X Y)^\perp, where (\cdot)^\perp denotes the orthogonal projection to the normal bundle N_pM. This bilinear form is symmetric and valued in the normal space, quantifying how geodesics on M deviate from straight lines in the ambient space. For example, on a sphere of radius r, \mathrm{II}(X, Y) = -\frac{1}{r} \langle X, Y \rangle \nu, where \nu is the outward unit normal, illustrating uniform bending. Associated with the second fundamental form is the shape operator (or Weingarten map), which describes the differential of the Gauss map into the normal space. For a unit normal vector field \nu along M, the shape operator S_\nu: T_pM \to T_pM is given by S_\nu X = -\nabla_X \nu, where the derivative is the ambient connection projected to the . This relates the normal variation to tangential changes; its eigenvalues are the principal curvatures, which indicate maximal and minimal bending directions. In the case of hypersurfaces (where \dim N_pM = 1), the shape operator fully encodes the extrinsic geometry via the second fundamental form through \mathrm{II}(X, Y) = \langle S_\nu X, Y \rangle. Key extrinsic invariants derived from the shape operator include the and . The vector \mathbf{H} for a is \mathbf{H} = \frac{1}{m} \mathrm{trace}(S_\nu) \nu, where m = \dim M, representing the average principal curvatures and pointing in the direction of minimal volume change for normal variations; in coordinates, H = \frac{1}{m} g^{ij} h_{ij} with h_{ij} = \langle \mathrm{II}(\partial_i, \partial_j), \nu \rangle. For surfaces in \mathbb{R}^3, the K, which coincides with the intrinsic sectional curvature by Gauss's , equals \det S_\nu = k_1 k_2, the product of principal curvatures, distinguishing elliptic (K > 0), parabolic (K = 0), and hyperbolic (K < 0) points. This equality highlights how extrinsic embeddings realize intrinsic metrics, with K serving as a baseline for comparing embedded geometries. The Codazzi-Mainardi equations provide compatibility conditions linking the second fundamental form to the intrinsic , ensuring the embedding is consistent. These equations state that the covariant derivative of \mathrm{II} is symmetric in its arguments: (\nabla_X \mathrm{II})(Y, Z) - (\nabla_Y \mathrm{II})(X, Z) = R(X, Y) Z^\perp, where R is the intrinsic curvature tensor, though for hypersurfaces it simplifies to \partial_i h_{jk} - \Gamma^l_{ij} h_{lk} - \Gamma^l_{ik} h_{jl} = 0 in coordinates (up to normal components). Originally derived by Mainardi in and Codazzi in 1860, these equations, alongside the Gauss equation R(X,Y)Z = (\mathrm{II}(Y,Z))^\top X - (\mathrm{II}(X,Z))^\top Y (projected to tangent), govern the integrability of systems defining immersions. They are crucial for proving local existence of embeddings with prescribed metric and second fundamental form. A cornerstone result in extrinsic geometry is the , which asserts that any (M, g) admits a local into \mathbb{R}^N for sufficiently large N. Proved by in using a argument on the space of immersions, the theorem guarantees C^\infty embeddings for compact manifolds into \mathbb{R}^{n(n+1)(3n+11)/2} (where n = \dim M) and global embeddings for non-compact cases under additional conditions. This resolves the embedding problem by showing that intrinsic metrics can always be realized extrinsically, with applications to rigidity and approximation of abstract geometries.

Intrinsic versus extrinsic comparison

In differential geometry, the intrinsic and extrinsic approaches to studying manifolds offer complementary perspectives, with key equivalences emerging in specific cases while revealing fundamental limitations in others. A seminal equivalence for surfaces arises from Gauss's , which demonstrates that the , typically expressed extrinsically through the second fundamental form, is intrinsically determined by the alone, independent of any embedding in ambient space. This result establishes that certain extrinsic quantities can be recovered intrinsically, allowing local geometric properties to be analyzed without reference to an external . Despite such local equivalences, extrinsic methods face significant global limitations, as not all intrinsic geometries admit isometric embeddings into . For instance, Hilbert's theorem proves that the hyperbolic plane, with its constant negative , cannot be isometrically immersed as a complete surface into three-dimensional , highlighting the impossibility of realizing certain intrinsic metrics extrinsically on a global scale. This limitation underscores the challenges of extrinsic approaches in capturing the full and of non-Euclidean manifolds without distortions. The intrinsic approach proves advantageous in higher dimensions, where constructing explicit embeddings becomes computationally infeasible or theoretically intractable, enabling the study of abstract Riemannian structures solely through internal metrics and curvatures. Conversely, extrinsic excels in and , particularly for low-dimensional cases like surfaces in \mathbb{R}^3, where embeddings facilitate geometric computations and illustrations. These trade-offs guide the choice of method: intrinsic for generality and high-dimensional analysis, extrinsic for concrete, embeddable models. Weyl's embedding problem further illustrates the interplay, posing the question of whether a given on the sphere with nonnegative can be realized as the induced on a surface in , with affirmative solutions under certain smoothness conditions. In modern developments, particularly in conformal geometry, syntheses of intrinsic and extrinsic viewpoints employ tractor bundles—associated bundles over a manifold equipped with a canonical connection—to construct that preserve conformal structure without relying on a fixed ambient , offering a unified framework for both local and global analyses.

Applications

In general relativity and physics

In general relativity, spacetime is modeled as a four-dimensional Lorentzian manifold, a smooth pseudo-Riemannian manifold equipped with a metric tensor of signature (3,1) or (1,3) that allows for the distinction between timelike, spacelike, and null intervals, enabling the description of causal structure and light cones. This framework, developed by Albert Einstein, replaces the flat Minkowski spacetime of special relativity with a curved geometry where the metric satisfies the Einstein field equations, which relate the curvature of spacetime to the distribution of mass and energy: R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \frac{8\pi G}{c^4} T_{\mu\nu}, where R_{\mu\nu} is the Ricci curvature tensor, R is the scalar curvature, g_{\mu\nu} is the metric tensor, T_{\mu\nu} is the stress-energy tensor, G is the gravitational constant, and c is the speed of light. These equations, first presented in their final form in Einstein's 1915-1916 papers, govern the dynamics of gravity as geometric curvature induced by matter and energy. Exact solutions to the Einstein equations provide models for specific gravitational phenomena, such as black holes. The describes the geometry outside a spherically symmetric, non-rotating M, given by ds^2 = -\left(1 - \frac{2GM}{c^2 r}\right) c^2 dt^2 + \left(1 - \frac{2GM}{c^2 r}\right)^{-1} dr^2 + r^2 d\Omega^2, where d\Omega^2 = d\theta^2 + \sin^2\theta \, d\phi^2 is the metric on the unit sphere; this solution, derived shortly after the field equations, reveals features like the event horizon at the r_s = 2GM/c^2. Perturbations around such backgrounds lead to , which propagate as ripples in . In the weak-field limit, these are described by the linearized Einstein equations, where the metric perturbation h_{\mu\nu} satisfies a \square \bar{h}_{\mu\nu} = -16\pi G T_{\mu\nu}/c^4 in the harmonic gauge, predicting transverse-traceless waves traveling at the ; Einstein first derived this in , with refinements in 1918 for energy flux. Differential geometry extends to quantum gravity theories, notably , where the ten-dimensional includes compactified often modeled as Calabi-Yau manifolds—complex Kähler manifolds with vanishing first and Ricci-flat metrics—to preserve and yield effective four-dimensional physics. These manifolds, whose existence was proven by Yau in 1978, allow to accommodate the standard model's three generations of fermions through topological invariants like and Hodge numbers. More recently, posits that the information content of a volume of space is encoded on its boundary, as in the AdS/CFT , which equates type IIB on times a five-sphere to a on the boundary; proposed by Maldacena in 1997, this duality has advanced understanding of and strongly coupled systems since the . The principle originated in 't Hooft's 1993 work on entropy and was formalized by Susskind in 1995.

In computer science and graphics

Differential geometry plays a pivotal role in and through discrete differential geometry (DDG), which approximates continuous geometric concepts on discrete structures like triangle meshes to enable efficient algorithms for modeling, simulation, and analysis. DDG provides tools for processing surfaces in applications such as , where smooth deformations require curvature-aware operations, and , where geodesic paths guide on non-Euclidean terrains. In , DDG-inspired methods facilitate shape analysis and manifold learning by discretizing Riemannian structures for computational tractability. A core concept in is discrete curvature on triangle meshes, which quantifies local bending analogous to in smooth surfaces. The cotangent formula computes the integrated K_i at vertex i as K_i = \frac{1}{A_i} \sum_{j \in \mathcal{N}(i)} (\cot \alpha_{ij} + \cot \beta_{ij}), where A_i is the Voronoi area around vertex i, \mathcal{N}(i) are neighboring vertices, and \alpha_{ij}, \beta_{ij} are opposite angles in the triangles adjacent to edge ij. This formula arises from the cotangent Laplacian operator and approximates the continuous via angle defects, enabling applications like surface fairing and feature detection in models. Discrete geodesics, the shortest paths on mesh surfaces respecting intrinsic distances, are computed using methods like the heat method or fast marching. The heat method solves the on the mesh to propagate distances from a source, yielding geodesic distances in linear time relative to mesh size by leveraging the cotangent Laplacian for . It initializes a from the source points, normalizes the result to obtain distance fields, and projects onto the gradient of the , making it robust for complex topologies like genus-g surfaces in . Fast marching, an alternative, advances a front of known distances across the mesh using Dijkstra-like on unfolded triangles, providing geodesics with sub-quadratic for moderate meshes. Shape analysis in graphics employs the Gromov-Hausdorff (GH) distance to compare manifolds intrinsically, measuring the minimal distortion needed to isometrically embed one into the other via a correspondence. The GH distance d_{GH}(X, Y) is defined as \inf \{ d_H(f(X), g(Y)) \mid f: X \to Z, g: Y \to Z \text{ isometric embeddings into some metric space } Z \}, where d_H is the ; this metric quantifies shape similarity under deformations, invariant to rigid motions. This metric supports tasks like non-rigid registration in , where aligning scanned models requires handling partial isometries, and has been approximated via embeddings or for efficient computation on high-resolution scans. DDG underpins practical applications such as subdivision surfaces, which refine coarse meshes into smooth limits while preserving geometric properties like . Subdivision exterior calculus extends DDG operators to hierarchical Catmull-Clark surfaces, enabling the solution of partial differential equations for and deformation in pipelines. In games, DDG aids by precomputing discrete curvatures and geodesics to cull non-intersecting regions, accelerating broad-phase tests for dynamic meshes in real-time simulations like character interactions. Recent advances integrate with , where neural networks learn Riemannian metrics on data manifolds to capture non-Euclidean geometries in high-dimensional spaces. Post-2010 works show that deep networks implicitly parameterize metrics, aligning data distributions via learned coordinate systems that approximate distances on underlying manifolds. For instance, Riemannian residual networks extend ResNets to manifolds by evolving features along geodesics, improving tasks like shape classification in graphics datasets.

In other sciences and engineering

Differential geometry finds significant applications in , where the Fisher-Rao metric provides a Riemannian structure on the space of probability distributions, enabling the analysis of statistical models through geometric tools. This metric, defined on a manifold of probability densities p(x|\theta) parameterized by coordinates \theta^i, takes the form ds^2 = \sum_{i,j} g_{ij} \, d\theta^i \, d\theta^j, where g_{ij}(\theta) = \int (\partial_i \log p)(\partial_j \log p) \, p(x|\theta) \, dx, and it quantifies the distinguishability between nearby distributions in terms of . Introduced in the context of statistical estimation, this metric allows for the study of geodesics and curvatures that correspond to natural gradients in optimization problems, such as on exponential families. In applications, it facilitates the geometric interpretation of divergences like the Kullback-Leibler, promoting insights into algorithms and training dynamics. In biology, differential geometry models the shapes of elastic structures like DNA using the Kirchhoff rod theory, which describes the equilibrium configurations of thin, inextensible rods under bending, twisting, and stretching energies. The Kirchhoff equations, derived from variational principles, govern the dynamics and statics of these rods, capturing supercoiling phenomena where DNA twists upon itself to minimize elastic energy. For instance, in modeling supercoiled DNA plasmids, the theory predicts plectonemic and toroidal writhe formations, aligning with experimental observations of topological linking numbers in bacterial chromosomes. This approach integrates extrinsic curvature (bending) and intrinsic twist, providing a framework for simulating nucleosome wrapping and chromatin folding, with parameters calibrated to DNA's persistence length of approximately 50 nm. Control theory employs sub-Riemannian geometry to address nonholonomic systems, where constraints limit allowable velocities, leading to a metric defined only on a distribution of the tangent space rather than the full bundle. In such systems, geodesics represent optimal paths under differential constraints, as seen in the car-parking problem, where a vehicle with fixed wheel direction must maneuver into a tight space via sequences of forward and reverse motions. This example, modeled on the Heisenberg group, illustrates how sub-Riemannian distances grow quadratically in some directions, enabling motion planning algorithms that compute shortest paths in configuration spaces like SE(2) for mobile robots. The geometry underpins controllability via Chow's theorem, ensuring reachability in nilpotent approximations, with applications extending to underactuated mechanical systems in robotics. In , optimal transport theory on manifolds uses Wasserstein to quantify differences between distributions of resources or agents, facilitating models of allocation and matching under geometric constraints. The Wasserstein distance, a on the of probability measures endowed with a Riemannian structure, minimizes the cost of transporting mass along manifold , generalizing the case to curved like economic state . Seminal applications include in spatial economies, where the captures transportation costs on networks modeled as manifolds, leading to insights on Pareto optimality and of matching markets. For example, in labor economics, it analyzes wage distributions across skill manifolds, revealing how geometric bottlenecks affect measures. Engineering leverages conformal mappings from differential geometry for , transforming complex shapes into simpler domains to solve equations. The Joukowski transformation, z \mapsto z + \frac{1}{z}, maps circles to wing-like profiles while preserving angles, allowing for to be solved via uniform flow around the circle. This method predicts lift coefficients and stall behaviors for , as validated in early tests, and extends to multi-element wings via Schwarz-Christoffel mappings for polygonal boundaries. In modern design, it optimizes shapes for minimal drag, with the mapping's analyticity ensuring irrotational flow approximations hold for low Reynolds numbers.