The Chern-Gauss-Bonnet theorem on semi-Riemannian manifolds

08.11.2024, 15:00  –  2.09.2.22
Verteidigung Masterarbeit / Master's Thesis Defence

Marten Steuer (UP)

The Chern-Gauss-Bonnet theorem is one of the most fundamental results in Riemannian geometry, which connects the geometry of a compact, oriented Riemannian manifold \( (M,g)\) with its topology. For closed manifolds, this theorem yields the formula:
\[\chi(M)=\int_M \text{Pf}\left(\frac{\Omega}{2\pi}\right),\]
where \( \chi(M)\) is the Euler characteristic, and \( \text{Pf} \left(\frac{\Omega}{2\pi}\right)\) is the Pfaffian of the curvature form \( \Omega\) which represents the Euler class.

Since it is possible to define the integral in the above formula for semi-Riemannian manifolds, it is natural to ask whether it still equals the Euler characteristic of the manifold. We showed that with a suitable modification of the formula, one can generalize the Chern-Gauss-Bonnet theorem to semi-Riemannian manifolds. This is achieved using the technique of analytic continuation, which has the advantage of extending the theorem without requiring too much additional theory.