Peter Petersen - Riemannian Geometry

Springer International Publishing AG 2016

Peter Petersen 1

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In this chapter we introduce the spaces and maps that pervade the subject. Without discussing any theory we present several examples of basic Riemannian manifolds and Riemannian maps. All of these examples will be at the heart of future investigations into constructions of Riemannian manifolds with various interesting properties.

The abstract definition of a Riemannian manifold used today dates back only to the 1930s as it wasnt really until Whitneys work in 1936 that mathematicians obtained a clear understanding of what abstract manifolds were other than just being submanifolds of Euclidean space. Riemann himself defined Riemannian metrics only on domains in Euclidean space. Riemannian manifolds where then metric objects that locally looked like a Riemannian metric on a domain in Euclidean space. It is, however, important to realize that this local approach to a global theory of Riemannian manifolds is as honest as the modern top-down approach.

Prior to Riemann, other famous mathematicians such as Euler, Monge, and Gauss only worked with 2-dimensional curved geometry. Riemanns invention of multi-dimensional geometry is quite curious. The story goes that Gauss was on Riemanns defense committee for his Habilitation (doctorate). In those days, the candidate was asked to submit three topics in advance, with the implicit understanding that the committee would ask to hear about the first topic (the actual thesis was on Fourier series and the Riemann integral). Riemanns third topic was On the Hypotheses which lie at the Foundations of Geometry. Evidently, he was hoping that the committee would select from the first two topics, which were on material he had already developed. Gauss, however, always being in an inquisitive mood, decided he wanted to hear whether Riemann had anything to say about the subject on which he, Gauss, was the reigning expert. Thus, much to Riemanns dismay, he had to go home and invent Riemannian geometry to satisfy Gausss curiosity. No doubt Gauss was suitably impressed, apparently a very rare occurrence for him.

From Riemanns work it appears that he worked with changing metrics mostly by multiplying them by a function (conformal change). By conformally changing the standard Euclidean metric he was able to construct all three constant curvature geometries in one fell swoop for the first time ever. Soon after Riemanns discoveries it was realized that in polar coordinates one can change the metric in a different way, now referred to as a warped product. This also exhibits all constant curvature geometries in a unified way. Of course, Gauss already knew about polar coordinate representations on surfaces, and rotationally symmetric metrics were studied even earlier by Clairaut. But those examples are much simpler than the higher-dimensional analogues. Throughout this book we emphasize the importance of these special warped products and polar coordinates. It is not far to go from warped products to doubly warped products, which will also be defined in this chapter, but they dont seem to have attracted much attention until Schwarzschild discovered a vacuum space-time that wasnt flat. Since then, doubly warped products have been at the heart of many examples and counterexamples in Riemannian geometry.

Another important way of finding examples of Riemannian metrics is by using left-invariant metrics on Lie groups. This leads us, among other things, to the Hopf fibration and Berger spheres. Both of these are of fundamental importance and are also at the core of a large number of examples in Riemannian geometry. These will also be defined here and studied further throughout the book.