Gaussian curvature - Citizendia

From left to right: a surface of negative Gaussian curvature (hyperboloid), a surface of zero Gaussian curvature (cylinder), and a surface of positive Gaussian curvature (sphere). In Mathematics, a hyperboloid is a Quadric, a type of surface in three Dimensions described by the equation {x^2 \over a^2} + A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe

In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface is the product of the principal curvatures, κ₁ and κ₂, of the given point. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Differential geometry, the two principal curvatures at a given point of a Surface measure how the surface bends by different amounts in different directions It is an intrinsic measure of curvature, i. e. , its value depends only on how distances are measured on the surface, not on the way it is embedded in space. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group This result is the content of Gauss's Theorema egregium. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German Gauss's Theorema Egregium (Latin "Remarkable Theorem" is a foundational result in Differential geometry proved by Carl Friedrich Gauss that concerns the

Symbolically, the Gaussian curvature Κ is defined as

$\Kappa = \kappa_1 \kappa_2 \,\!$ . In Mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry

It is also given by

$\Kappa = \frac{\langle (\nabla_2 \nabla_1 - \nabla_1 \nabla_2)\mathbf{e}_1, \mathbf{e}_2\rangle}{\det g},$

where $\nabla_i = \nabla_{{\mathbf e}_i}$ is the covariant derivative and g is the metric tensor. In Mathematics, the covariant derivative is a way of specifying a Derivative along Tangent vectors of a Manifold. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space

At a point p on a regular surface in R³, the Gaussian curvature is also given by

$K(\mathbf{p}) = \det(S(\mathbf{p})),$

where S is the shape operator. This is a glossary of some terms used in Riemannian geometry and Metric geometry &mdash it doesn't cover the terminology of Differential topology.

A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates. For Liouville's equation in dynamical systems see Liouville's theorem (Hamiltonian. In Mathematics, specifically in Differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric

1 Informal definition
2 Total curvature
3 Important theorems
- 3.1 Theorema egregium
- 3.2 Gauss–Bonnet theorem
4 Surfaces of constant curvature
5 Alternative Formulas
6 References
7 See also

Informal definition

We represent the surface by the implicit function theorem as the graph of a function f of 2 variables, and assume the point p is a critical point, i. e. the gradient of f vanishes (this can always be attained by a suitable rigid motion). Then the Gaussian curvature of the surface at p is the determinant of the Hessian matrix of f, i. e. the 2 by 2 matrix of second derivatives. This definition allows one immediately to grasp the distinction between cup/cap versus saddle point behavior in terms of second year calculus.

Total curvature

The sum of the angles of a triangle on a surface of negative curvature is less than that of a plane triangle.

The surface integral of the Gaussian curvature over some region of a surface is called the total curvature. In Mathematics, a surface integral is a Definite integral taken over a Surface (which may be a curved set in Space) it can be thought The total curvature of a geodesic triangle equals the deviation of the sum of its angles from $π$ . In Mathematics, a geodesic /ˌdʒiəˈdɛsɪk -ˈdisɪk/ -dee-sik is a generalization of the notion of a " straight line " to " curved spaces A triangle is one of the basic Shapes of Geometry: a Polygon with three corners or vertices and three sides or edges which are Line The sum of the angles of a triangle on a surface of positive curvature will exceed $π$ , while the sum of the angles of a triangle on a surface of negative curvature will be less than $π$ . On a surface of zero curvature, such as the Euclidean plane, the angles will sum to precisely $π$ . Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria.

$\sum_{i=1}^3 \theta_i = \pi + \iint_T K \,dA.$

A more general result is the Gauss-Bonnet Theorem. The Gauss–Bonnet theorem or Gauss–Bonnet formula in Differential geometry is an important statement about Surfaces which connects their geometry (in

Important theorems

Theorema egregium

Main article: Theorema Egregium

Gauss's Theorema Egregium (Latin: "remarkable theorem") states that Gaussian curvature of a surface can be determined from the measurements of length on the surface itself. Gauss's Theorema Egregium (Latin "Remarkable Theorem" is a foundational result in Differential geometry proved by Carl Friedrich Gauss that concerns the In fact, it can be found given the full knowledge of the first fundamental form and expressed via the first fundamental form and its partial derivatives of first and second order. In Differential geometry, the first fundamental form is the Inner product on the Tangent space of a Surface in three-dimensional Euclidean In Mathematics, a partial derivative of a function of several variables is its Derivative with respect to one of those variables with the others held constant Equivalently, the determinant of the second fundamental form of a surface in R³ can be so expressed. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Differential geometry, the second fundamental form is a Quadratic form on the Tangent plane of a smooth surface in the three dimensional The "remarkable", and surprising, feature of this theorem is that although the definition of the Gaussian curvature of a surface S in R³ certainly depends on the way in which the surface is located in space, the end result, the Gaussian curvature itself, is determined by the inner metric of the surface without any further reference to the ambient space: it is an intrinsic invariant. The term intrinsic denotes a characteristic or property of some thing or action which is essential and specific to that thing or action and which is wholly independent In Mathematics, an invariant is something that does not change under a set of transformations The property of being an invariant is invariance. In particular, the Gaussian curvature is invariant under isometric deformations of the surface. For the Mechanical engineering and Architecture usage see Isometric projection.

In contemporary differential geometry, a "surface", viewed abstractly, is a two-dimensional differentiable manifold. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. To connect this point of view with the classical theory of surfaces, such an abstract surface is embedded into R³ and endowed with the Riemannian metric given by the first fundamental form. In Mathematics, the differential geometry of surfaces deals with smooth Surfaces with various additional structures most often a Riemannian metric In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M Suppose that the image of the embedding is a surface S in R³. A local isometry is a diffeomorphism f: U → V between open regions of R³ whose restriction to S ∩ U is an isometry onto its image. In Mathematics, a diffeomorphism is an Isomorphism of Smooth manifolds It is an Invertible function that maps one Differentiable For the Mechanical engineering and Architecture usage see Isometric projection. Theorema Egregium is then stated as follows:

The Gaussian curvature of an embedded smooth surface in R³ is invariant under the local isometries.

For example, the Gaussian curvature of a cylindrical tube is zero, the same as for the "unrolled" tube (which is flat). A cylinder is one of the most basic curvilinear geometric shapes the Surface formed by the points at a fixed distance from a given Straight line, the axis ^[1] On the other hand, since a sphere of radius R has constant positive curvature R⁻² and a flat plane has constant curvature 0, these two surfaces are not isometric, even locally. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe Thus any planar representation of even a part of a sphere must distort the distances. Therefore, no cartographic projection is perfect. A map projection is any method of representing the Surface of a sphere or other shape on a plane.

Gauss–Bonnet theorem

Main article: Gauss-Bonnet theorem

The Gauss-Bonnet theorem links the total curvature of a surface to its Euler characteristic and provides an important link between local geometric properties and global topological properties. The Gauss–Bonnet theorem or Gauss–Bonnet formula in Differential geometry is an important statement about Surfaces which connects their geometry (in In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant

Surfaces of constant curvature

Minding's theorem (1839) states that all surfaces with the same constant curvature K are locally isometric. For the Mechanical engineering and Architecture usage see Isometric projection. A consequence of Minding's theorem is that any surface whose curvature is identically zero can be constructed by bending some plane region. Such surfaces are called developable surfaces. In Mathematics, a developable surface is a Surface with zero Gaussian curvature. Minding also raised the question whether a closed surface with constant positive curvature is necessarily rigid. In Mathematics a closed surface (2- Manifold) is a Closed manifold of dimension two with a single Connected component.

Liebmann's theorem (1900) answered Minding's question. The only regular (of class C²) closed surfaces in R³ with constant positive Gaussian curvature are spheres. "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe ^[2]

Hilbert's theorem (1901) states that there exists no complete analytic (class C^ω) regular surface in R³ of constant negative Gaussian curvature. In Differential geometry, Hilbert's theorem (1901 states that there exists no complete regular surface S of constant negative Gaussian curvature K In fact, the conclusion also holds for surfaces of class C² immersed in R³, but breaks down for C¹-surfaces. The pseudosphere has constant negative Gaussian curvature except at its singular cusp. In Geometry, a pseudosphere of radius R is a surface of curvature &minus1/ R 2 (precisely a complete, Simply connected In Singularity theory a cusp is a singular point of a curve. Spinode is an alternative name but this is less commonly used today ^[3]

Alternative Formulas

Gaussian curvature of a surface in R³ can be expressed as the ratio of the determinants of the second and first fundamental forms:

$K = \frac{\det II}{\det I} = \frac{LN-M^2}{EG-F^2}.$

The Brioschi formula gives Gaussian curvature solely in terms of the first fundamental form:

$K = \frac{\begin{vmatrix} -\frac{1}{2}E_{vv} + F_{uv} - \frac{1}{2}G_{uu} & \frac{1}{2}E_u & F_u-\frac{1}{2}E_v\\F_v-\frac{1}{2}G_u & E & F\\\frac{1}{2}G_v & F & G \end{vmatrix}-\begin{vmatrix} 0 & \frac{1}{2}E_v & \frac{1}{2}G_u\\\frac{1}{2}E_v & E & F\\\frac{1}{2}G_u & F & G \end{vmatrix}} {(EG-F^2)^2}$

For an orthogonal parametrization, Gaussian curvature is:

$K = -\frac{1}{2\sqrt{EG}}\left(\frac{\partial}{\partial u}\frac{G_u}{\sqrt{EG}} + \frac{\partial}{\partial v}\frac{E_v}{\sqrt{EG}}\right).$

Gaussian curvature is the limiting difference between the circumference of a geodesic circle and a circle in the plane:

$K = \lim_{r \rarr 0} (2 \pi r - \mbox{C}(r)) \cdot \frac{3}{\pi r^3}$

Gaussian curvature is the limiting difference between the area of a geodesic circle and a circle in the plane:

$K = \lim_{r \rarr 0} (\pi r^2 - \mbox{A}(r)) \cdot \frac{12}{\pi r^4}$

Gaussian curvature may be expressed with the Christoffel symbols: ^[4]

$K = -\frac{1}{E} \left( \frac{\partial}{\partial u}\Gamma_{12}^2 - \frac{\partial}{\partial v}\Gamma_{11}^2 + \Gamma_{12}^1\Gamma_{11}^2 - \Gamma_{11}^1\Gamma_{12}^2 + \Gamma_{12}^2\Gamma_{12}^2 - \Gamma_{11}^2\Gamma_{22}^2\right)$

References

^ Porteous, I. In Algebra, a determinant is a function depending on n that associates a scalar, det( A) to every n × n In Differential geometry, the second fundamental form is a Quadratic form on the Tangent plane of a smooth surface in the three dimensional In Differential geometry, the first fundamental form is the Inner product on the Tangent space of a Surface in three-dimensional Euclidean In Mathematics, orthogonal coordinates are defined as a set of d coordinates q = ( q 1 q 2. The circumference is the distance around a closed Curve. Circumference is a kind of Perimeter. Area is a Quantity expressing the two- Dimensional size of a defined part of a Surface, typically a region bounded by a closed Curve. In Mathematics and Physics, the Christoffel symbols, named for Elwin Bruno Christoffel (1829–1900 are coordinate-space expressions for the Levi-Civita R. , Geometric Differentiation. Cambridge University Press, 1994. ISBN 0-521-39063-X
^ Kühnel, Wolfgang (2006). Differential Geometry: Curves - Surfaces - Manifolds. American Mathematical Society. ISBN 0821839888.
^ Hilbert theorem. Springer Online Reference Works.
^ Struik, Dirk (1988). Lectures on Classical Differential Geometry. Courier Dover Publications. ISBN 0486656098.

Contents