This lecture is an introduction to differential geometry, more precisely to semi-Riemannian manifolds, their curvature and global properties.<br> The main topic are Riemannian metrics on manifolds. The simplest examples are surfaces in Euclidean space ℝ<sup>3</sup>. Such surfaces may be intrinsically curved, as e.g. the sphere. Or they may only be extrinsically curved, as e.g. a cylinder -- which may be cut by a "scissor" and then this surface is isometric to an open set of a plane.<br>The goal is to understand not only surfaces, but similar curvature quantities in arbitrary dimensions and codimensions, a generalization going back to work of Bernhard Riemann. Very similar structures were later used by Einstein and others in order to get a mathematical framework to describe general relativity.The theory is still a very active area in mathematics and theoretical physics.
Gemeinsames Seminar von Prof. Bernd Ammann (Mathematik) und Gunnar Bali (Physik)
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Let M be a compact connected smooth manifold without boundary. One of the guiding questions of the lecture is whether there is a Riemannian metric g on M, such that g has everywhere positive scalar curvature. We will shortly call such a metric a psc (positive scalar curvature) metric.
The question actually splitts into two parts: 1.) Obstructions against psc metrics, in other words: reasons why such metrics cannot exist 2.) Constructions of psc metrics on large classes of manifolds
A large part of the lecture will study the first part: The Atiyah-Singer index theorem is the most important obstruction against psc metrics. The theorem has attracted a lot of interest within mathematics, because it has connected many fields in mathematics: geometry, topology, and partial differential equations. It also established many links to applications in mathematical physics. For example it provides important tools for a better understanding of scalar curvature in general relativity, leading e.g. to Witten's proof of the positive mass theorem of an asymptotically Euclidean spacetime (e.g. a star or a black hole). The index theorem has attracted many important prizes, e.g. the Fields Medal for Atiyah in 1966 and the Abel prize for Atiyah and Singer in 2004. The Atiyah-Singer index theorem states that the Fredholm index of an elliptic partial differential operator D on M is equal to a characteristic class of the tangent bundle of M, integrated over M. In the classical case, D is the Dirac operator and if M carries a psc metric, then this Fredholm index is 0. On the other hand, characteristic classes are easy to calculate, they do not depend on the choice of a Riemannian metric, and often we see that they are not zero. As a consequence we get manifolds that do not carry a psc metric. The Atiyah-Singer theorem also applies to other types of elliptic operators. One special case is the Gaus-Bonnet-Chern operator which yields a higher-dimensional version of the Gauss-Bonnet formula, and in another version we obtain as an index the signature that some people in the audience might have seen in a topology course. We want to follow the heat-kernel method to prove the index theorem. This approach is considerably simpler that the original approach by Atiyah and Singer, and allows us to understand the proof in many details. If time permits we will then study the second part of the question and we will use surgery methods to construct many metrics of positive scalar curvature.
A good a impression about the course can be obtained from the book(s) by Roe or the lecture notes cited on the web page.
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Im zweiten Teil der Vorlesung entwickeln wir Techniken, um Vektorbündel voneinander zu unterscheiden (charakteristische Klassen), und wir wollen allgemein die Topologie von Faser-Bündeln betrachten.