The Mathematical Seminar: On curvature and the Theorem of Gauss-Bonnet

Abstract: In this seminar, we discuss the notion of curvature of a surface in the 3-dimensional Euclidean space. As the name suggests, curvatures describes how much the surface "bends" in the ambient space, and it is a local invariant of the embedded surface. The concept of curvature has many important application in mathematics and physics such as in the theory of general relativity and gauge theories. While the curvature depends on the particular choice of embedding of the surface, the Theorem of Gauss-Bonnet asserts that the average curvature is independent of the embedding and only depends on the topology of the surface. This theorem can for instance be used to compute the sum of the angles of a triangle on a sphere and to prohibit certain embeddings of given surface into the Euclidean space.
Starting from curves in surfaces and then going to surfaces in 3-space, I will give an intuitive introduction to curvature. After that I will explain the Theorem of Gauss-Bonnet and some of its consequences.

The mathematics seminar is a seminar series for students at the bachelor's programme in mathematics. All interested are also welcome to participate.