“Gauss-Bonnet Formula”的概念、定义、习题解题方法及翻译-数学辞典

单词

Gauss-Bonnet Formula

释义

Gauss-Bonnet Formula

The Gauss-Bonnet formula has several formulations. The simplest one expresses the total Gaussian Curvature of anembedded triangle in terms of the total Geodesic Curvature of the boundary and the JumpAngles at the corners.

More specifically, if is any 2-D Riemannian Manifold (like asurface in 3-space) and if is an embedded triangle, then the Gauss-Bonnet formula states that the integral over thewhole triangle of the Gaussian Curvature with respect to Area is given by minus the sum of theJump Angles minus the integral of the Geodesic Curvature over the whole of the boundary ofthe triangle (with respect to Arc Length),

(1)

where

is the Gaussian Curvature,

is the Area measure, the

s are the Jump Angles of

, and

is the Geodesic Curvature of

, with

the Arc Lengthmeasure.

The next most common formulation of the Gauss-Bonnet formula is that for any compact, boundaryless 2-DRiemannian Manifold, the integral of the Gaussian Curvature over the entire Manifold with respect toArea is times the Euler Characteristic of the Manifold,

(2)

This is somewhat surprising because the total Gaussian Curvature is differential-geometric in character, but theEuler Characteristic is topological in character and does not depend on differential geometry at all. So if youdistort the surface and change the curvature at any location, regardless of how you do it, the same total curvature ismaintained.

Another way of looking at the Gauss-Bonnet theorem for surfaces in 3-space is that the Gauss Map of the surface hasDegree given by half the Euler Characteristic of the surface

(3)

which works only for Orientable Surfaces. This makes the Gauss-Bonnet theorem a simpleconsequence of the Poincare-Hopf Index Theorem, which is a nice way of looking at things if you're a topologist, but notso nice for a differential geometer. This proof can be found in Guillemin and Pollack (1974). Millman and Parker (1977) givea standard differential-geometric proof of the Gauss-Bonnet theorem, and Singer and Thorpe (1996) give a Gauss's TheoremaEgregium-inspired proof which is entirely intrinsic, without any reference to the ambient Euclidean Space.

A general Gauss-Bonnet formula that takes into account both formulas can also be given. For any compact 2-DRiemannian Manifold with corners, the integral of the Gaussian Curvature over the 2-Manifold withrespect to Area is times the Euler Characteristic of the Manifold minus the sum of theJump Angles and the total Geodesic Curvature of the boundary.

References

Chavel, I. Riemannian Geometry: A Modern Introduction. New York: Cambridge University Press, 1994.

Guillemin, V. and Pollack, A. Differential Topology. Englewood Cliffs, NJ: Prentice-Hall, 1974.

Millman, R. S. and Parker, G. D. Elements of Differential Geometry. Prentice-Hall, 1977.

Reckziegel, H. In Mathematical Models from the Collections of Universities and Museums (Ed. G. Fischer). Braunschweig, Germany: Vieweg, p. 31, 1986.

Singer, I. M. and Thorpe, J. A. Lecture Notes on Elementary Topology and Geometry. New York: Springer-Verlag, 1996.

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