ABSTRACT
As we have already stated, Differential Geometry is, essentially, nothing else but the study of curvature. Indeed, we made curvature the red thread running along through this book. In fact, one might even say that if one has to chose a leitmotif for our book, then one might say that the “story” behind this text is one of the development of the notion of curvature in all its facets. Thus it is only natural, given that, this book, as any else in this world, must, after all, be finite, to concentrate in these concluding chapters on curvature in higher dimensions, in its many and various facets. We shall see that, as expected (or feared) higher dimensional generalizations are necessarily more complicated. We shall also learn however, that, most surprisingly, some of the more complicated notions are precisely the ones that lend themselves to discretization, simplification and dimensionality reduction. Given that it is not possible to give, in the confines of this book, a proper, formal derivation of curvature in higher dimension, perhaps the main reason for introducing them is precisely the fact that they allow to open a practically new world of modern, discrete Differential Geometry with a variety of novel and important applications. Thus we restrict all the technical details, that properly belong to a Riemannian Geometry book, to a modicum that allows us to proceed as fast as possible (yet still coherently) toward the geometric core intended.
