A few areas of science the Gauss-Bonnet Theorem is used in.
GRAVITATIONAL
LENSING
In this geometrical approach to gravitational lensing theory, we apply the Gauss-Bonnet theorem to the optical metric of a lens, modelled as a static, spherically symmetric, perfect non-relativistic fluid, in the weak deflection limit.
The rotation of the plane of polarization of light in an optical fibre whose initial and final directions are parallel is given by the Berry phase, which results from geometrical considerations, by appealing to the Gauss-Bonnet theorem in the tangent space of the fibre
The Gauss-Bonnet theorem for a polyhedron (a union of finitely many compact convex polytopes) in n-dimensional Euclidean space expresses the Euler characteristic of the polyhedron as a sum of certain curvatures, which are different from zero only at the vertices of the polyhedron.
