Titlebook: Riemannian Geometry; Peter Petersen Textbook 19981st edition Springer Science+Business Media New York 1998 Riemannian geometry.Spinor.Tens

POINT

排名真古怪

制度

下边深陷

Firefly

Ricci Curvature Comparison,developed: relative volume comparison and weak upper bounds for the Laplacian of distance functions. With these techniques we shall show numerous results on restrictions of fundamental groups of such spaces and also present a different proof of the estimate for the first Betti number by Bochner. The

spondylosis

Convergence, a sequence of Riemannian manifolds, or more generally metric spaces, to converge to a space. In the first section we develop the weakest convergence concept: Gromov-Hausdorff convergence. We then go on to explain some of the elliptic regularity theory we need for some of the later developments. We

灰心丧气

Sectional Curvature Comparison II,is critical point technique is used in the proofs of all the big theorems in this chapter. The other important technique comes from Toponogov’s theorem, which we prove in the next section. The first applications of these new ideas are to sphere theorems. We then prove the soul theorem of Cheeger and

利用

翅膀拍动

Geodesics and Distance,are smooth and therefore show the existence of the kind of distance functions we worked with earlier. In the last section we give some metric characterizations of Riemannian isometries and submersions.

吹气

Symmetric Spaces and Holonomy,paces are related Finally, we classify all compact manifolds with nonnegative curvature operator. We shall in a few places use results from Chapter 9. They will therefore have to be taken for granted at this point.