- Gaussian curvature
differential geometry, the Gaussian curvature or Gauss curvature of a point on a surfaceis the product of the principal curvatures, "κ"1 and "κ"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 in space. This result is the content of Gauss's Theorema egregium.
Symbolically, the Gaussian
curvature"Κ" is defined as:.
It is also given by: where is the
covariant derivativeand "g" is the metric tensor.
At a point p on a regular surface in R"3", the Gaussian curvature is also given by: where "S" is the
A useful formula for the Gaussian curvature is Liouville's equation in terms of the Laplacian in
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.
surface integralof the Gaussian curvature over some region of a surface is called the total curvature. The total curvature of a geodesic triangleequals 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'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 formand expressed via the first fundamental form and its partial derivatives of first and second order. Equivalently, the determinantof the second fundamental formof 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 intrinsicinvariant. In particular, the Gaussian curvature is invariant under isometric deformations of the surface.
differential geometry, a "surface", viewed abstractly, is a two-dimensional differentiable manifold. 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 metricgiven by the first fundamental form. 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 isometryonto 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 tube is zero, the same as for the "unrolled" tube (which is flat). [Porteous, I. R., "Geometric Differentiation". Cambridge University Press, 1994. ISBN 0-521-39063-X] On the other hand, since a
sphereof 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 projectionis perfect.
The Gauss-Bonnet theorem links the total curvature of a surface to its
Euler characteristicand 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. 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. Minding also raised the question whether a closed surfacewith constant positive curvature is necessarily rigid.
*Liebmann's theorem (1900) answered Minding's question. The only regular (of class "C"2) closed surfaces in R3 with constant positive Gaussian curvature are
spheres. [cite book | last = Kühnel | first = Wolfgang | title = Differential Geometry: Curves - Surfaces - Manifolds | publisher = American Mathematical Society | date = 2006 | id = ISBN 0821839888]
* Hilbert's theorem (1901) states that there exists no complete analytic (class "C""ω") regular surface in R3 of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class "C"2 immersed in R3, but breaks down for "C"1-surfaces. The
pseudospherehas constant negative Gaussian curvature except at its singular cusp. [ [http://eom.springer.de/h/h047410.htm "Hilbert theorem". Springer Online Reference Works.] ]
*The Brioschi formula gives Gaussian curvature solely in terms of the first fundamental form::
*For an orthogonal parametrization, Gaussian curvature is::
*Gaussian curvature is the limiting difference between the
circumferenceof a geodesic circle and a circle in the plane::
*Gaussian curvature is the limiting difference between the
areaof a geodesic circle and a circle in the plane::
*Gaussian curvature may be expressed with the
Christoffel symbols: [cite book | last = Struik | first = Dirk| title = Lectures on Classical Differential Geometry | publisher = Courier DoverPublications | date = 1988 | id = ISBN 0486656098] :
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