Gauss–Bonnet theorem

The Gauss -Bonnet (after Carl Friedrich Gauss and Pierre Ossian Bonnet ) is an important statement about surfaces which connects their geometry with its topology by a relationship between curvature and Euler characteristic is produced. This rate was found independent of both mathematicians.

Statement

Let be a compact orientable two-dimensional Riemannian and manifold with boundary. Denote by the Gaußkrümmung in points by and with the geodesic curvature of the boundary curve. Then we have

The Euler characteristic of is. The kit can be applied to manifolds without boundary in particular. Then the term is omitted.

If an area is, the set can also be formulated for piecewise differentiable boundary curves. In this case, it appears on the left side an additional term:

The exterior angle AW are defined as the angle between the right - and left-side limit of the tangent at the inflection points of. The edge curve shall be oriented so as to show to the surface. Here, the normal vector of the surface and the tangent to the edge curve.

One can generalize the theorem of Gauss -Bonnet on simplicial surfaces, we define the angle defect of a corner as discrete Gaußkrümmung.

Statement of the theorem

Distorted to the manifold, then its Euler characteristic remains unchanged, in contrast to Gaußkrümmung at each point. The theorem states that the integral of the curvature, so the total curvature remains unchanged.

Examples

The round sphere of radius 1 has at each point of the Gaussian curvature 1 The integral of the Gaussian curvature corresponds to their surface. On the other hand, the Euler Characteristic 2, because it receives (ie 2-1 1 = 2), the sphere as the bonding of two ( rounded ) surface along an edge of a corner.

Theorema elegantissimum

This coming from Gaussian inference states that the total curvature of a simply connected geodesic triangle is equal to the angle excess. For the special case of the 2-sphere can be seen on the exterior angle sum of an infinitesimal (ie flat ) triangle of the equivalence to the Gauss -Bonnet. The equivalence applies, however - in the two-dimensional case - also in general, which can be viewed using a triangulation, because this applies to:

Gauss -Bonnet- Chern

The theorem can be generalized to dimensions that was done by André Weil and Carl B. Allendoerfer 1943 and with new evidence by Chern, 1944.

Let be a compact oriented Riemannian manifold with even dimension and be the Riemannian curvature tensor. As is true of this, this can be used as vector-valued differential form

Be understood. Under these conditions shall be considered

The Pfaffian determinant.

With the knowledge that applies to the Fredholm index of the equality, the exterior derivative is, this sentence can be understood as a special case of the Atiyah-Singer index theorem. In this context, the set of Gaussian Bonnet Chern thus provides a means for calculating the index of the topological operator.