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Gaussian curvature
 
 
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In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 is the product of the principal curvature
Principal curvature

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 at that point....
s, ?1 and ?2, of the given point. 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
Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
 in space. This result is the content of Gauss's
Carl Friedrich Gauss

Johann Carl Friedrich Gauss. was a Germans mathematician and scientist who contributed significantly to many fields, including number theory, statistics, mathematical analysis, Differential geometry and topology, geodesy, electrostatics, astronomy and optics....
 Theorema egregium
Theorema Egregium

Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces....
.

Symbolically, the Gaussian curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
 ? is defined as .

It is also given by
where is the covariant derivative
Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a connection on the frame bundle &mdas...
 and g is the metric tensor
Metric tensor

In the mathematics field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of Vector in Euclidean space....
.

At a point p on a regular surface in R3, the Gaussian curvature is also given by
where S is the shape operator.

A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates
Isothermal coordinates

In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifoldare local coordinates where the metric tensor is...
.

Informal definition
We represent the surface by the implicit function theorem as the graph of a function, f, of two 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).






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In differential geometry, the Gaussian curvature or Gauss curvature of a point on a surface
Surface

In mathematics, specifically in topology, a surface is a two-dimensional topological manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space E3....
 is the product of the principal curvature
Principal curvature

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 at that point....
s, ?1 and ?2, of the given point. 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
Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
 in space. This result is the content of Gauss's
Carl Friedrich Gauss

Johann Carl Friedrich Gauss. was a Germans mathematician and scientist who contributed significantly to many fields, including number theory, statistics, mathematical analysis, Differential geometry and topology, geodesy, electrostatics, astronomy and optics....
 Theorema egregium
Theorema Egregium

Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces....
.

Symbolically, the Gaussian curvature
Curvature

In mathematics, curvature refers to any of a number of loosely related concepts in different areas of geometry. Intuitively, curvature is the amount by which a geometric object deviates from being flat, or straight in the case of a line , but this is defined in different ways depending on the context....
 ? is defined as .

It is also given by
where is the covariant derivative
Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a connection on the frame bundle &mdas...
 and g is the metric tensor
Metric tensor

In the mathematics field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of Vector in Euclidean space....
.

At a point p on a regular surface in R3, the Gaussian curvature is also given by
where S is the shape operator.

A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in isothermal coordinates
Isothermal coordinates

In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifoldare local coordinates where the metric tensor is...
.

Informal definition


We represent the surface by the implicit function theorem as the graph of a function, f, of two 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
Hessian matrix

In mathematics, the Hessian matrix is the square matrix of second-order partial derivatives of a function ; that is, it describes the local curvature of a function of many variables....
 of f (being the products of the eigenvalues of the Hessian). (Recall that the Hessian is 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 surface integral
Surface integral

In mathematics, a surface integral is a definite integral taken over a surface ; it can be thought of as the double integral analog of the line integral....
 of the Gaussian curvature over some region of a surface is called the total curvature. The total curvature of a geodesic
Geodesic

In mathematics, a geodesic [jee-uh-des-ik, -dee-sik] is a generalization of the notion of a "Line " to "manifolds".In presence of a Metric , geodesics are defined to be the shortest path between points on the space....
 triangle
Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or wikt:vertex and three sides or edges which are line segments....
 equals the deviation of the sum of its angles from . The sum of the angles of a triangle on a surface of positive curvature will exceed , while the sum of the angles of a triangle on a surface of negative curvature will be less than . On a surface of zero curvature, such as the Euclidean plane, the angles will sum to precisely .

A more general result is the Gauss-Bonnet Theorem.

Important theorems


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. In fact, it can be found given the full knowledge of the first fundamental form
First fundamental form

In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3....
 and expressed via the first fundamental form and its partial derivative
Partial derivative

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 ....
s of first and second order. Equivalently, the determinant
Determinant

In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
 of the second fundamental form
Second fundamental form

In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a differential geometry of surfaces in the three dimensional Euclidean space, usually denoted by II....
 of a surface in R3 can be so expressed. 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
Invariant (mathematics)

In mathematics, an invariant is something that does not change under a set of Transformation s. The property of being an invariant is invariance....
. In particular, the Gaussian curvature is invariant under isometric
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
 deformations of the surface.

In contemporary differential geometry, a "surface", viewed abstractly, is a two-dimensional differentiable manifold
Differentiable manifold

A differentiable manifold is a type of manifold that is locally similar enough to Euclidean space to allow one to do calculus. This article deals with differentiability in different contexts including: smooth function, k times differentiable, and holomorphic function....
. To connect this point of view with the classical theory of surfaces
Differential geometry of surfaces

In mathematics, the differential geometry of surfaces deals with smooth manifold surfaces with various additional structures, most often, a Riemannian metric....
, such an abstract surface is embedded
Embedding

In mathematics, an embedding is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup....
 into R3 and endowed with the Riemannian metric given by the first fundamental form. Suppose that the image of the embedding is a surface S in R3. A local isometry is a diffeomorphism
Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that map s one differentiable manifold to another, such that both the function and its inverse are smooth function....
 f: U → V between open regions of R3 whose restriction to S ∩ U is an isometry
Isometry

In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
 onto its image. 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
Cylinder (geometry)

A cylinder is one of the most curvilinear basic geometric shapes: the surface formed by the points at a fixed distance from a given straight line, the axis of the cylinder....
 tube is zero, the same as for the "unrolled" tube (which is flat). On the other hand, since a sphere
Sphere

A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
 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. Thus any planar representation of even a part of a sphere must distort the distances. Therefore, no cartographic projection is perfect.

Gauss–Bonnet theorem

The Gauss-Bonnet theorem links the total curvature of a surface to its Euler characteristic
Euler characteristic

In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent....
 and provides an important link between local geometric properties and global topological properties.

Surfaces of constant curvature


  • Minding's theorem (1839) states that all surfaces with the same constant curvature K are locally isometric
    Isometry

    In mathematics, an isometry, isometric isomorphism or congruence mapping is a distance-preserving isomorphism between metric spaces....
    . 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 surface
    Developable surface

    In mathematics, a developable surface is a surface with zero Gaussian curvature. That is, it is "surface" that can be Flatness onto a Plane without distortion ....
    s. Minding also raised the question whether a closed surface
    Closed surface

    In mathematics a closed surface is a closed manifold of dimension two, with a single connected component. Examples are spaces like the sphere, the torus, and the Klein bottle....
     with constant positive curvature is necessarily rigid.


  • Liebmann's theorem (1900) answered Minding's question. The only regular (of class C2) closed surfaces in R3 with constant positive Gaussian curvature are sphere
    Sphere

    A sphere is a symmetrical geometrical object. In non-mathematical usage, the term is used to refer either to a round ball or to its two-dimensional surface....
    s.


  • Hilbert's theorem
    Hilbert's theorem (differential geometry)

    In differential geometry, Hilbert's theorem states that there exists no complete regular surface of constant negative Gaussian curvature immersed in ....
     (1901) states that there exists no complete analytic (class Cω) regular surface in R3 of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class C2 immersed in R3, but breaks down for C1-surfaces. The pseudosphere
    Pseudosphere

    In geometry, a pseudosphere of radius R is a surface of curvature −1/R2 , by analogy with the sphere of radius R, which is a surface of curvature 1/R2....
     has constant negative Gaussian curvature except at its singular cusp
    Cusp (singularity)

    In singularity theory a cusp is a Singular point of a curve. Spinode is an alternative name, but this is less commonly used today.For a curve defined as the zero set of a function of two variables , the cusps on the curve will have the following properties:...
    .


Alternative Formulas


  • Gaussian curvature of a surface in R3 can be expressed as the ratio of the determinant
    Determinant

    In algebra, a determinant is a function depending on n that associates a scalar , det, to an n?n square matrix A. The fundamental geometric meaning of a determinant is a scale factor for measure when A is regarded as a linear transformation....
    s of the second
    Second fundamental form

    In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a differential geometry of surfaces in the three dimensional Euclidean space, usually denoted by II....
     and first
    First fundamental form

    In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3....
     fundamental forms:


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


  • For an orthogonal
    Orthogonal coordinates

    In mathematics, orthogonal coordinates are defined as a set of d coordinates q = in which the coordinate surfaces all meet at right angles ....
     parametrization    , Gaussian curvature is:


  • Gaussian curvature is the limiting difference between the circumference
    Circumference

    The circumference is the distance around a closed curve. Circumference is a kind of perimeter....
     of a geodesic circle     and a circle in the plane:


  • Gaussian curvature is the limiting difference between the area
    Area

    Area is a quantity expressing the two-dimensional size of a defined part of a surface, typically a region bounded by a closed curve. The term surface area refers to the total area of the exposed surface of a 3-dimensional solid, such as the sum of the areas of the exposed sides of a polyhedron....
     of a geodesic circle     and a circle in the plane:


  • Gaussian curvature may be expressed with the Christoffel symbols
    Christoffel symbols

    In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor....
    :


See also

  • sectional curvature
    Sectional curvature

    In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature depends on a two-dimensional plane in the tangent space at p....
  • Mean curvature
    Mean curvature

    In mathematics, the mean curvature of a surface is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedding surface in some ambient space such as Euclidean space....
  • Theorema egregium
    Theorema Egregium

    Gauss's Theorema Egregium is a foundational result in differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces....
  • Gauss map
    Gauss map

    In differential geometry, the Gauss map maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: X ? S2 such that N is a unit vector orthogonal to X at p, namely the normal ve...