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Gauss map

In differential geometry, the Gauss map (or Gaussian map) of an oriented surface S in Euclidean three-dimensional space \mathbb{R}^3 is a smooth function N: S \to S^2 that assigns to each point p \in S the unit normal vector N(p) at that point, thereby mapping the surface to the unit sphere S^2.^[1]^[2] Named after the mathematician Carl Friedrich Gauss, the concept originates from his foundational 1827 work Disquisitiones generales circa superficies curvas, where he developed the theory of surface curvatures using normal vectors, though the explicit term "Gauss map" arose later in modern treatments.^[3]^[4] The Gauss map encodes essential geometric information about the surface, particularly through its differential dN_p: T_p S \to T_{N(p)} S^2, which (via the natural identification of tangent spaces) is an endomorphism of the tangent space at p and whose eigenvalues are the principal curvatures of S at p.^[1] The Gaussian curvature K(p) at a point p is defined as the determinant of this differential, K(p) = \det(dN_p), measuring the intrinsic deviation of the surface from being flat, while the mean curvature H(p) is given by H(p) = -\frac{1}{2} \operatorname{tr}(dN_p), capturing extrinsic bending.^[5]^[2] For a parametrized surface with first fundamental form coefficients and unit normal \vec{N}, the absolute Gaussian curvature satisfies |K(p)| = \frac{\| \vec{N}_u \times \vec{N}_v \|}{\| \vec{X}_u \times \vec{X}_v \|}, which equals the Jacobian of the Gauss map, linking local curvature to the map's stretching.^[2] A key application of the Gauss map is the Gauss-Bonnet theorem, which for a closed orientable surface S states that the integral of the Gaussian curvature equals $2\pi times the Euler characteristic: \iint_S K \, dA = 2\pi \chi(S). This integral also equals $4\pi times the degree of the Gauss map, revealing how total curvature relates to the surface's global topology.^[6]^[2] For orientable surfaces, the existence of a continuous Gauss map defines the orientation, and the map's degree (as a map to S^2) equals half the Euler characteristic of the surface, providing a topological invariant.^[1] In higher dimensions or for hypersurfaces in \mathbb{R}^n, generalizations of the Gauss map extend these ideas to study degenerate cases and focal sets, with applications in computer graphics, robotics, and the analysis of minimal surfaces. The Weingarten map, which is -dN_p, further linearizes the normal variations and is central to shape operator computations in surface theory.^[7]

Definition and Fundamentals

Definition for Surfaces

The Gauss map of an oriented regular surface S in Euclidean 3-space \mathbb{R}^3 is defined as the smooth map N: S \to S^2 that sends each point p \in S to the unit normal vector N(p) perpendicular to the tangent plane T_p S, where S^2 denotes the unit sphere in \mathbb{R}^3.^[1] This vector N(p) is chosen to be consistent with the given orientation of S, ensuring that the map is well-defined and continuous across the surface.^[2] The orientation of S is crucial for the construction of the Gauss map, as the tangent plane at each point admits two possible unit normals pointing in opposite directions; the orientation specifies a consistent choice, often via a nowhere-vanishing normal vector field.^[8] In local coordinates, if \sigma: U \subset \mathbb{R}^2 \to S is a parametrization of S, the unit normal is given explicitly by

N(u,v) = \frac{\sigma_u(u,v) \times \sigma_v(u,v)}{\| \sigma_u(u,v) \times \sigma_v(u,v) \|},

where \sigma_u and \sigma_v are the partial derivatives with respect to the parameters u and v, and the cross product aligns with the orientation.^[2] A simple example is the unit sphere S^2 = \{ x \in \mathbb{R}^3 : \|x\| = 1 \} with the outward orientation, for which the Gauss map coincides with the identity map N(p) = p for all p \in S^2, since the position vector at each point is already the unit outward normal.^[9] This illustrates how the Gauss map encodes the geometric orientation of the surface onto the sphere of directions.^[10]

Historical Context

The concept of the Gauss map was introduced by Carl Friedrich Gauss in his seminal 1827 paper "Disquisitiones generales circa superficies curvas," where it arose as a tool for investigating the curvature properties of surfaces embedded in three-dimensional Euclidean space.^[3] In this work, Gauss employed the Gauss map to formulate his theorema egregium, a groundbreaking result that connects the intrinsic geometry of a surface—measurable without reference to the ambient space—to its extrinsic curvature via the determinant of the differential of the map.^[11] The explicit term "Gauss map" was introduced later by Wilhelm Blaschke in his early 20th-century textbooks on differential geometry.^[12] During the late 19th century, Gaston Darboux and other geometers expanded on these foundations, examining the Gauss map's properties for particular surface classes such as quadrics, as detailed in Darboux's multi-volume treatise "Leçons sur la théorie générale des surfaces" (1887–1896).^[13] The Gauss map saw renewed interest in the mid-20th century within the study of minimal surfaces and harmonic mappings, with Robert Osserman's 1960s analyses highlighting its role in characterizing complete minimal surfaces of finite total curvature through the map's image on the unit sphere.^[14]

Local Properties

Differential of the Gauss Map

The differential of the Gauss map at a point p on an oriented surface S \subset \mathbb{R}^3, denoted dN_p, is the linear map dN_p: T_p S \to T_{N(p)} S^2, where N: S \to S^2 is the Gauss map assigning the unit normal vector at each point.^[15]^[16] Since T_{N(p)} S^2 is the plane orthogonal to N(p) in \mathbb{R}^3, and T_p S shares this orthogonality, dN_p identifies with an endomorphism of T_p S.^[15]^[16] The shape operator, or Weingarten map, is the self-adjoint linear operator S_p = -dN_p: T_p S \to T_p S, which measures the variation of the tangent plane along the surface.^[15]^[16] For tangent vectors X, Y \in T_p S, the second fundamental form satisfies II_p(X, Y) = \langle S_p X, Y \rangle = -\langle dN_p X, Y \rangle, where \langle \cdot, \cdot \rangle denotes the Euclidean inner product induced on T_p S.^[15]^[16] The principal curvatures k_1(p) and k_2(p) at p are the eigenvalues of S_p, corresponding to orthogonal principal directions in T_p S that extremize the normal curvature.^[15]^[16] In local coordinates (u, v) parametrizing S, with first fundamental form matrix I = \begin{pmatrix} E & F \\ F & G \end{pmatrix} and second fundamental form matrix II = \begin{pmatrix} e & f \\ f & g \end{pmatrix}, the matrix representation of S_p is I^{-1} II.^[15] For a plane, the Gauss map N is constant, so dN_p = 0, the shape operator S_p is the zero map, and both principal curvatures vanish.^[15]^[16] For a right circular cylinder of radius r, parametrized by height z and angle \theta, the Gauss map N(\theta, z) = (\cos \theta, \sin \theta, 0) yields dN_p = 0 along the generator direction (partial with respect to z), so S_p has eigenvalues 0 and $1/r, with the nonzero eigenvalue along the circumferential direction.^[15]^[16]

Relation to Gaussian Curvature

The Gaussian curvature K(p) at a point p on an oriented surface S \subset \mathbb{R}^3 is given by the determinant of the differential of the Gauss map dN_p: T_p S \to T_{N(p)} S^2, where the tangent spaces are identified via the induced metric, so K(p) = \det(dN_p).^[7] To derive this relation, consider a local parametrization X(u,v) of S near p, with coordinate basis \{X_u, X_v\}. The differential dN_p acts on this basis, yielding dN_p(X_u) = N_u and dN_p(X_v) = N_v, where N is the unit normal from the Gauss map. The matrix representation of dN_p in this basis is -I^{-1} II, where I = \begin{pmatrix} E & F \\ F & G \end{pmatrix} and II = \begin{pmatrix} e & f \\ f & g \end{pmatrix} are the matrices of the first and second fundamental forms. Thus, \det(dN_p) = \det(-I^{-1} II) = \det(II)/\det(I) = (eg - f^2)/(EG - F^2) = K, matching the standard formula for Gaussian curvature and the Jacobian determinant of N with respect to the induced metric.^[7] For oriented surfaces, the sign of K follows the orientation convention: K > 0 at elliptic points where the surface bends similarly in all directions (like a sphere), K < 0 at hyperbolic points with saddle-like bending, and K = 0 at parabolic points.^[2]^[7] A representative example is the unit sphere S^2, where the Gauss map N is the identity map, so dN_p is the identity linear map on each tangent space, yielding \det(dN_p) = 1 everywhere, consistent with the constant Gaussian curvature K = 1.^[17] This pointwise relation implies Gauss's theorema egregium, which states that K depends only on the intrinsic metric of S (first fundamental form), even though the Gauss map N and its differential are defined extrinsically via the embedding in \mathbb{R}^3.^[7]

Global Properties

Total Curvature

The total curvature of an oriented surface S \subset \mathbb{R}^3 is defined as the surface integral \int_S K \, dA, where K is the Gaussian curvature of S and dA is its area element. This integral equals the signed area swept out by the image of the Gauss map N: S \to S^2.^[10] For a compact, closed, oriented surface S, the Gauss-Bonnet theorem states that the total curvature satisfies \int_S K \, dA = 2\pi \chi(S), where \chi(S) denotes the Euler characteristic of S.^[18] This topological invariant relates the global geometry of S to the degree of its Gauss map, given by \deg(N) = \frac{1}{4\pi} \int_S K \, dA.^[7] For the standard embedding of the 2-sphere S^2 of radius 1, the Gaussian curvature is constantly 1, yielding total curvature $4\pi and \deg(N) = 1.^[7] In contrast, the standard torus of revolution has total curvature 0 and \deg(N) = 0, consistent with its Euler characteristic of 0.^[7] A consequence of degree theory for the Gauss map, related to Hopf's theorem on the index formula for vector fields, is the inequality |\deg(N)| \leq \frac{1}{4\pi} \int_S |K| \, dA, bounding the topological degree by the total absolute curvature.^[19]

Degree of the Gauss Map

The topological degree of the Gauss map N: S \to S^2 for a compact oriented surface S embedded in \mathbb{R}^3 is the integer \deg(N) that counts the signed number of preimages of a regular value of N, where the sign is determined by whether the orientation induced by N at the preimage matches that of S^2. This degree is a homotopy invariant, remaining unchanged under continuous deformations of the embedding that preserve orientation, and thus serves as a topological invariant useful for classifying surfaces up to homeomorphism. For smooth closed orientable surfaces, the degree relates directly to the total Gaussian curvature via \deg(N) = \frac{1}{4\pi} \int_S K \, dA, where K is the Gaussian curvature; this follows from the fact that the pullback of the volume form on S^2 under N equals K \, dA, and the integral of the volume form over S^2 is $4\pi. By the Gauss-Bonnet theorem, this also equals half the Euler characteristic \chi(S)/2. Representative examples illustrate this invariant for non-compact minimal surfaces with finite total curvature, where the Gauss map extends meromorphically to the compactification of the parameter domain. For Enneper's surface, a complete immersed minimal surface of genus zero, \deg(N) = -1, corresponding to its total curvature of -4\pi. The catenoid, another complete minimal surface with two ends, has total curvature of -4\pi and \deg(N) = -1.^[20] For surfaces of higher genus, the degree can be computed using algebraic topology: \deg(N) is the image of the fundamental class [S] \in H_2(S; \mathbb{Z}) under the induced homomorphism N_*: H_2(S; \mathbb{Z}) \to H_2(S^2; \mathbb{Z}) \cong \mathbb{Z}, or dually via cohomology as the integer such that N^*([\omega]) = \deg(N) [\omega], where [\omega] is the generator of H^2(S^2; \mathbb{Z}) given by the volume form.

Singularities

Cusps

In the context of the Gauss map for a surface in Euclidean 3-space, a cusp singularity arises in the image when the map exhibits a fold along a ridge curve that intersects a parabolic line on the surface, resulting in a sharp, pointed feature on the unit sphere where tangent planes from nearby points converge.^[21] These cusps typically occur at parabolic points, where the Gaussian curvature vanishes (K=0), and the differential of the Gauss map has rank less than 2, causing the image to fail to be an immersion.^[22] Geometrically, along a cusp ridge—often a line of extremal curvature—the surface maintains zero Gaussian curvature, leading to a folding of the Gauss map that produces the cusp. For generic smooth surfaces, such cusps form precisely at parabolic points that serve as limits of inflection (flex) points along asymptotic curves, where the asymptotic direction aligns with the parabolic curve.^[21] This configuration results in three topologically distinct patterns of asymptotic lines near the cusp: a hyperbolic type and two elliptic types, distinguished by the behavior of the principal curvature directions relative to the parabolic boundary.^[22] On ruled or developable surfaces, where Gaussian curvature is identically zero, the Gauss map degenerates to a curve on the unit sphere, as the unit normal remains constant along each ruling. Cusps appear on this spherical curve at director directions corresponding to the striction curve, where the rulings achieve minimal length or maximal twist. For instance, the tangent developable surface generated by the tangent lines to a space curve features cusps in its Gauss map image along the spherical indicatrix of the edge of regression, marking points where the surface folds sharply.^[21] In contrast, a right circular cylinder maps smoothly to a great circle without cusps, while a general cone or tangent developable introduces cusps due to varying ruling orientations.^[22] The nature of these cusp singularities can be analyzed using the Dupin indicatrix, which at parabolic points degenerates into a pair of parallel lines in the tangent plane, reflecting the semi-cubic parabola shape of the local Gauss map image and indicating an A_2-type singularity. Alternatively, the binormal curvature of the associated asymptotic curves provides a measure of the cusp's sharpness, vanishing at the flex points that limit to the parabolic locus and confirming the fold's geometric severity.^[22] Visualizations of the Gauss map for ruled surfaces often depict this image as a closed spherical curve punctuated by cusps, highlighting the directions of the rulings that contribute to the surface's developability.^[21]

Other Singular Points

In addition to cusps, the Gauss map of a surface in Euclidean 3-space exhibits other types of singular points, including flat points. Flat points occur where both principal curvatures vanish, resulting in zero Gaussian curvature and a degenerate differential of the Gauss map.^[23] Swallowtail singularities represent higher-order phenomena in generic immersions, classified within singularity theory as A₃-type points where the map germ is equivalent to the unfolding f(x,y) = (x, y^3 + x y). These arise along the parabolic set and indicate transitions in the topology of the Gauss image.^[24] The Darboux classification provides a framework for generic umbilic points on surfaces, dividing them into three types—lemon, monstar, and star—based on the local configuration of lines of curvature emanating from the point. An illustrative example is the monkey saddle surface, defined parametrically as \mathbf{r}(u,v) = (u, v, u^3 - 3uv^2), which features a flat umbilic at the origin where the Gauss map collapses to a point with higher multiplicity.^[25]^[7] Local models for these singularities often involve Taylor expansions of the Gauss map \mathbf{N} near the critical point. For instance, at a generic umbilic, the expansion reveals transverse intersections in the image, such as \mathbf{N}(u,v) = \mathbf{N}_0 + a(u^2 - v^2) \mathbf{e}_1 + 2b uv \mathbf{e}_2 + \ higher\ terms, where the quadratic form captures the isotropic nature of the differential.^[7] These singular points impact global invariants like total curvature, as the standard degree of the Gauss map may fail; instead, generalized notions, such as the topological degree accounting for branch points, are required to relate the integral of Gaussian curvature to the enclosed solid angle on the unit sphere.^[26]

Generalizations

To Hypersurfaces

For a hypersurface M^{n-1} immersed in Euclidean space \mathbb{R}^n, the Gauss map N: M \to S^{n-1} is defined by assigning to each point p \in M the unit normal vector n(p) spanning the one-dimensional normal space to M at p.^[27] This map generalizes the classical Gauss map for surfaces in \mathbb{R}^3, where n=3.^[27] The differential dN_p: T_p M \to T_p M of the Gauss map at p identifies with a linear endomorphism of the tangent space, since the tangent space to the sphere at n(p) is orthogonal to n(p) and thus parallel to T_p M.^[27] The eigenvalues of dN_p, known as the principal curvatures \kappa_1(p), \dots, \kappa_{n-1}(p), determine the local bending along principal directions.^[27] The shape operator (or Weingarten map) S_p: T_p M \to T_p M is defined as S_p = -dN_p. The Gauss-Kronecker curvature K(p) at p is the determinant \det(dN_p) = \prod_{i=1}^{n-1} \kappa_i(p), which reduces to the Gaussian curvature when n=3.^[27] A representative example is the standard unit sphere S^{n-1} \subset \mathbb{R}^n, where the Gauss map N coincides with the identity map on S^{n-1}, and the differential dN_p is the identity endomorphism at each p, yielding all principal curvatures equal to 1 and K(p) = 1.^[27] For a compact oriented hypersurface without boundary, the total curvature integral \int_M \det(dN) \, dV_g, where dV_g is the induced volume element on M, equals the (n-1)-dimensional volume of S^{n-1} multiplied by the degree \deg(N) of the Gauss map.^[28] This relation follows from the fact that the pullback under N of the volume form on S^{n-1} yields \det(dN) \, dV_g, and the degree measures the oriented covering multiplicity.^[28]

To Higher Codimension Immersions

For an isometric immersion f: M^m \to \mathbb{R}^{m+k} with codimension k > 1, the Gauss map generalizes the classical construction by associating to each point x \in M the tangent m-plane f_*(T_x M) \subset T_{f(x)} \mathbb{R}^{m+k} \cong \mathbb{R}^{m+k}, yielding a map \nu: M \to \mathrm{Gr}(m, m+k) into the Grassmannian of m-dimensional subspaces of \mathbb{R}^{m+k}. This contrasts with the hypersurface case (k=1), where the target is the unit sphere via the normal direction. For oriented immersions, the Gauss map lifts to the Stiefel manifold V_m(\mathbb{R}^{m+k}) of orthonormal m-frames tangent to M, capturing the oriented tangent structure more precisely.^[29]^[30] The second fundamental form \alpha_f: TM \times TM \to NM is a smooth, symmetric \mathbb{R}^{m+k}-valued bilinear form on M, defined by \alpha_f(X, Y) = (\overline{\nabla}_X df(Y))^\perp for tangent vectors X, Y \in T_x M, where \overline{\nabla} is the flat connection on \mathbb{R}^{m+k} and ^\perp denotes the normal component. In higher codimension, \alpha_f decomposes into components along an orthonormal normal frame \{\xi_1, \dots, \xi_k\}, giving rise to shape operators A_{\xi_r}: TM \to TM via \langle \alpha_f(X, Y), \xi_r \rangle = \langle A_{\xi_r} X, Y \rangle, which measure bending in each normal direction. These operators satisfy the Codazzi equation (\overline{\nabla}_X A_{\xi_r}) Y - (\overline{\nabla}_Y A_{\xi_r}) X = A_{(\overline{\nabla}_X \xi_r)^\top} Y - A_{(\overline{\nabla}_Y \xi_r)^\top} X + R^\perp(X, Y) \xi_r, linking the extrinsic geometry to the normal connection.^[29]^[31] Curvature measures arise as traces and norms of these operators; the mean curvature vector is the \mathbb{R}^{m+k}-valued form H = \frac{1}{m} \sum_{i=1}^m \alpha_f(e_i, e_i) for an orthonormal tangent frame \{e_i\}, while higher analogs include the scalar curvature \mathrm{Scal}(M) = \sum_{r=1}^k \|A_{\xi_r}\|^2 - |\tau|^2, where \tau is the mean curvature in the normal bundle, derived from the Gauss equation R(X, Y) Z = (A_{\alpha_f(Y, Z)} X - A_{\alpha_f(X, Z)} Y). These quantify extrinsic invariants beyond the single Gaussian curvature of hypersurfaces. A representative example occurs for curves (m=1, k = n-1) immersed in \mathbb{R}^n, where the Gauss map is the unit tangent indicatrix \tau: I \to S^{n-1}, and the second fundamental form reduces to the curvature vector \kappa = \frac{d\tau}{ds}, with mean curvature |\kappa| tracing the curve's bending.^[29]^[31] In submanifold theory, the Gauss map facilitates rigidity results, such as extensions of the Ruh-Vilms theorem: a Gauss map into the Grassmannian is harmonic if and only if the mean curvature vector is parallel in the normal bundle. Applications appear in Cartan geometry, where adapted orthonormal frames along the immersion relate the Gauss map to the Maurer-Cartan structure equations for local equivalence of submanifolds. For instance, in the study of Legendrian submanifolds of codimension m+1 in \mathbb{R}^{2m+1} with the standard contact structure, the Gauss map to the Grassmannian encodes the contact distribution and aids in analyzing wavefront propagations and singularities.^[32]^[33]

References

[1]
Gauss Map -- from Wolfram MathWorld
The Gauss map is a function N from an oriented surface M in Euclidean space R^3 to the unit sphere in R^3. It associates to every point on the surface its ...
[2]
None
### Summary of the Gauss Map from the Document
[3]
[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 ...
[4]
Gauss Map - an overview | ScienceDirect Topics
The Gauss map is defined as a mathematical function that assigns to each point on a surface a unit normal vector at that point, providing important ...
[5]
https://mathworld.wolfram.com/GaussianCurvature.html
[6]
[PDF] 3. THE GEOMETRY OF THE GAUSS MAP - Penn Math
Feb 7, 2012 · Principal curvatures of the surface S at the point p . Lines of curvature on a surface. Definition of Gaussian curvature and mean curvature.
[7]
[PDF] 1 The Differential Geometry of Surfaces
Definition 7 Let S ⊂ R3 be a 2-dimensional surface with orientation N. The map N : S →. R3 taking values in the unit sphere S2 is called the Gauss map of S. 4 ...<|control11|><|separator|>
[8]
[PDF] M462 (HANDOUT 5) 0.1. Gauss Map. Definition 0.2. A regular ...
Definition 0.3. Given an oriented surface S with unit normal field N(p), we can regard this as a map from S to the unit sphere S2 = {(x, y, z) | x2 + y2 + z2 = ...
[9]
[PDF] The Gauss map
May 3, 2012 · For example, the unit sphere is covered by six parameterizations z = p1 − x2 − y2, y = √. 1 − x2 − z2, x = p1 − y2 − z2, z = −p1 ...
[10]
[PDF] General investigations of curved surfaces of 1827 and 1825 ...
original draft written out in 1825, but not published until 1900. A list of the editions and translations of the Paper of 1827 follows. There are three editions.
[11]
Leçons sur la théorie générale des surfaces et les applications ...
Apr 28, 2006 · Leçons sur la théorie générale des surfaces et les applications géométriques du calcul infinitésimal. by: Darboux, Gaston ... 1887-96. Topics ...Missing: map | Show results with:map
[12]
On the Gauss Curvature of Minimal Surfaces - jstor
This allows us to translate elementary properties of analytic functions into geometric statements about minimal surfaces. The variable w is obtained simply by ...
[13]
None
Below is a merged and comprehensive summary of the Gauss Map and related concepts from Manfredo P. do Carmo’s *Differential Geometry of Curves and Surfaces*, consolidating all information from the provided segments. To retain maximum detail, I will use a structured format with tables where appropriate, followed by a narrative summary for examples and additional context. The response avoids any "thinking" tokens as per the system constraint and focuses solely on presenting the merged information.
[14]
[PDF] DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces
directions are eigenvectors of the shape operator, the image of P under the Gauss map is nearly a small curvilinear rectangle at n.P / 2 † with sides k1h1 ...
[15]
[PDF] Gaussian and Mean Curvatures∗
The. Gauss map is bijective (one-to-one and onto) since every point on the ellipsoid has a distinct normal. The image region covers the unit sphere. Because the ...
[16]
[PDF] 7. THE GAUSS-BONNET THEOREM - Penn Math
Mar 29, 2012 · Since the total Gaussian curvature of a region on a surface in 3-space is the area of its Gauss image on S. 2. , we see that the total ...
[17]
[PDF] total curvature of complete submanifolds of euclidean space
The extrinsic version is closely related with the Hopf index theorem, with the mapping degree of the Gauss map and with the study of critical points of ...
[18]
Degree of the Gauss map and curvature integrals for closed ... - arXiv
Sep 15, 2016 · This application allows us to exhibit a list of topological invariants which combines the second fundamental form of the hypersurface and the ...Missing: surfaces | Show results with:surfaces
[19]
Cusps of the projective Gauss map
**Summary of Cusps of the Projective Gauss Map (McCrory & Shifrin, 1984)**
[20]
Cusps of Gauss Mappings: Characterizations of Gaussian Cusps
For a generic immersion X , there are three topologically distinct configurations of asymptotic lines near a cusp of the Gauss map (see figures 3.1, 3.2, 3.3).
[21]
[PDF] Geometry of Umbilics on Surfaces in the Three-space
Definition 1 A point of a surface is called umbilical, if the two principal curvatures at this point are equal. Recall that principal curvatures k1 and k2 ...Missing: vertex | Show results with:vertex
[22]
[PDF] Introduction to Differential Geometry of Space Curves and Surfaces
Dec 20, 2024 · The present book is about differential geometry of space curves and surfaces. The for- mulation and presentation are largely based on a ...
[23]
[PDF] Flat surfaces along swallowtails - arXiv
Sep 18, 2017 · For the study of singularities of such developable surfaces, we introduce the notion of Darboux frames along swallowtails and invariants. As a ...
[24]
Simple umbilics revisited - ScienceDirect.com
An umbilical point is called simple if it cannot be broken into simpler umbilical points by a small deformation of the corresponding surface in E 3 .
[25]
ON THE GAUSS MAP AND TOTAL CURVATURE OF COMPLETE ...
Our object in this paper is to put into final form the connection between total curvature and the Gauss map of complete minimal surfaces in R3 and.
[26]
[PDF] Pedal foliations and Gauss maps of hypersurfaces in Euclidean space
Abstract. The singular point of the Gauss map of a hypersurface in Euclidean space is the parabolic point where the Gauss-Kronecker curvature vanishes.
[27]
ON HYPERSURFACES OF LIE GROUPS - Project Euclid
O. Introduction. In his work on the theory of surfaces Gauss introduced what is called today the (normal) Gauss map of an orientable hypersurface in.
[28]
[PDF] SUBMANIFOLD THEORY BEYOND AN INTRODUCTION - Luis Florit
Submanifold theory has emerged as a natural development of the classical study of curves and surfaces in Euclidean three space with the methods of ...
[29]
[PDF] Gauss Maps and Symmetric Spaces
Sep 7, 2007 · ... Gauss map is the m- sphere which has the same dimension as M. In higher codimension we always have dim M < dim Gr, but sometimes the values ...
[30]
[PDF] riemannian submanifolds
Jul 7, 2013 · submanifolds of higher codimension: If f Mn → Sn+p (n ≥ 3,p > 1) is a full minimal immersion such that the shape operator Aξ in any normal.
[31]
Harmonic Gauss maps of submanifolds of arbitrary codimension of ...
Mar 1, 2023 · In this paper, we try to extend the codimension 1 case of Ruh–Vilms theorem to submanifolds M of arbitrary codimension by finding geometric ...
[32]
REAL KAEHLER SUBMANIFOLDS AND UNIQUENESS OF THE ...
It will turn out that locally, all isometric immersions with the same Gauss map can be described in terms of circular submanifolds, whereas globally, additional.