# Riemannian Geometry and Geometric Analysis (Universitext)

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## Riemannian geometry and geometric analysis - Jürgen Jost - Google Books

It enabled the formulation of Einstein 's general theory of relativity , made profound impact on group theory and representation theory , as well as analysis , and spurred the development of algebraic and differential topology. Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of non-Euclidean geometry. Every smooth manifold admits a Riemannian metric , which often helps to solve problems of differential topology.

It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds , which in four dimensions are the main objects of the theory of general relativity. Other generalizations of Riemannian geometry include Finsler geometry. There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and disclinations produce torsions and curvature. What follows is an incomplete list of the most classical theorems in Riemannian geometry.

The choice is made depending on its importance and elegance of formulation. Most of the results can be found in the classic monograph by Jeff Cheeger and D. Ebin see below. The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about.

## Riemannian geometry and geometric analysis

In all of the following theorems we assume some local behavior of the space usually formulated using curvature assumption to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at "sufficiently large" distances. From Wikipedia, the free encyclopedia.

Elliptic geometry is also sometimes called "Riemannian geometry". Projecting a sphere to a plane. Outline History. Therefore, the entire current structure of Riemannian geometry is summarized in this chapter, with complete references. There is probably no such summary anywhere else in the textbook literature.

Frankly, if this was the only chapter the book contained, it would still be worth having. With these results in hand, Chapter 5 gives a very compete and up to date treatment of the complex projective space; constructing Kahler manifolds and symmetric spaces as important examples of Riemannian manifolds Their geometry and associated Lie algebras are constructed in full detail, first in the compact case and then in the noncompact case. The noncompact case is developed in full as an example of a class of nonpositively curved Riemannian manifolds. The usual topics are discussed with more than usual clarity: saddle points on manifolds, nondegeneracy of critical points, the Morse Lemma, stable and unstable manifolds, foliations and more.

There are also more modern topics discussed such as graph flows and general orientations for manifolds at local extrema.

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It is also the first place in the book where infinite dimensional manifolds are briefly mentioned and defined. The main differences between them and the finite dimensional case are discussed in preparation for the Palais-Smale compactness condition and proving the existence of closed geodesics. Chapter 7 discusses the behavior of harmonic maps on general Riemannian manifolds by means of the energy density of local coordinates at a point on such manifolds.

This is a mainstream topic of geometric analysis; one of its main tools the Hamilton flow. This includes the higher regularity estimates and the generalized Bochner technique as well as the behavior of such maps on manifolds of nonpositive sectional curvature. Chapter 8 extends these results to multivariable harmonic maps defined on Riemann surfaces.

The book closes in a blaze of glory with a chapter on variational problems in quantum field theory. The machinery developed in the last four chapters is brought to bear on three of the most important functionals in string theory: The Ginzburg-Landau functional, the Seiberg-Witten functional and the class of Dirac-harmonic maps. These are developed as purely mathematical objects, but Jost is careful to point out differences in notation and the close relationship to physical models such as the nonlinear super-symmetric sigma model.

The amount of material Jost succeeds in covering in pages is positively astonishing, and he manages to do it completely coherently. Obviously this results in a book which is incredibly dense. The book is broken into bite-sized pieces, which helps a lot. Even so, the pace of the book can only be described as relentless.

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Each chapter builds on the ones before it, so that the chapters are never separate entities. These broaden the vision of the text even further with brief digressions into the history and current state of the central topic as well as exhaustive references for further study. As a result, the book actually ends up covering a lot more then it seems to, making it that much more impressive.

One part of the book that will appeal most to readers is that Jost gives lots of examples for every concept. Early in the book, these are given separately in batches of a dozen or so. As the concepts become more sophisticated, as in the variational analysis chapters, theorems are proven first in a special case and then if possible, generalized.

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This really helps with difficult concepts. I have just two minor complaints and a concern. The only place where pictures are plentiful is in the chapter on Morse theory. He is probably right; even so, Jost should have put a few more pictures in the book, particularly in the early chapters when most of the examples are easily visualizable. My other complaint is that there are not enough exercises. This seems to be a general problem with European textbooks. This is where my concern comes in. Can the book be used by itself for a graduate differential geometry course?

### 13 editions of this work

With a good teacher, any graduate student can use it to learn a lot of modern geometry very quickly. The problem is that the book has a very clear agenda and it has to focus totally on this agenda. But since Jost is so diligent about making his students familiar with the literature, choosing supplementary material will not be difficult at all.