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

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

(1) |

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

(2) |

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

(3) |

A general Gauss-Bonnet formula that takes into account both formulas can also be given. For any compact 2-D Riemannian Manifold with corners, the integral of the Gaussian Curvature over the 2-Manifold with respect to Area is times the Euler Characteristic of the Manifold minus the sum of the Jump 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.

© 1996-9

1999-05-25