Comparison theorems in riemannian geometry jeff cheeger, david g. Abstract the localization technique from convex geometry is generalized to the setting of riemannian manifolds whose ricci curvature is bounded from below. This process is experimental and the keywords may be updated as the learning algorithm improves. Cheeger, jeff, kleiner, bruce, and naor, assaf, acta mathematica, 2011. Aim of this book is to give a fairly complete treatment of the foundations of riemannian geometry through the tangent bundle and the geodesic flow on it. The uniformization theorem is a generalization of the riemann mapping theorem from proper simply connected open subsets of the plane to arbitrary simply connected riemann surfaces. Problems of traces 69 chapter 3 background material 1. Geometry is a rich source of opportunities for developing notions of proof. Contents 1 introduction 2 2 pdes in probability and geometry. Meanvalue theorems for riemannian manifolds proceedings. This gives, in particular, local notions of angle, length of curves, surface area and volume.
This second edition, first published in 2006, has a clearer treatment of many topics than the first edition, with new proofs of some theorems and a new chapter on the riemannian geometry of surfaces. Different from the riemannian case, these coordinates are not suitable for studying. In euclidean geometry, for example, two parallel lines. Riemannian manifold sectional curvature comparison theorem complete riemannian manifold parallel field these keywords were added by machine and not by the authors. This is a surv ey on the con v ergence theory dev elop ed rst b y cheeger and gromo v. Riemannian geometry is the branch of differential geometry that studies riemannian manifolds, smooth manifolds with a riemannian metric, i. This is a course on general relativity, given to part iii i. It covers advanced material, but is designed to be understandable for students who havent had a first course in the subject. We prove the existence and regularity of conformal harmonic co ordinates. This book is meant to be an introduction to riemannian geometry. Some remarks on the theory of spherical functions on symmetric riemannian manifolds. Professor jian song this dissertation consists of three parts, the rst one is on the blowup behavior of k ahler ricci ow on cpnblownup at one point, and the second one on the convergence of k ahler ricci. Deturckkazdan, some regularity theorems in riemannian geometry, 1981 greenewu, lipschitz convergence of riemannian manifolds, 1988 supplimental notes, with practice problems instructor.
These concern structure theories for manifolds satisfying bounds on sectional or ricci curvature, and related theories of geometric convergence. Riemannian geometry as determined by the volume of small geodesic balls. Introduction the subject of these lecture notes is comparison theory in riemannian geometry. Comparison geometry msri publications v olume 30, 1997 con v ergence theorems in riemannian geometry peter petersen abstra ct. Brian weber, brweber at math dot upenn dot edu office. Time permitting, penroses incompleteness theorems of general relativity will also be discussed. Some l p rigidity results for complete manifolds with harmonic curvature. Some sharp isoperimetric theorems for riemannian manifolds article pdf available in indiana university mathematics journal 493 september 2000 with 99 reads how we measure reads. Proof of the existence and uniqueness of geodesics. A beautiful mukhometovs theorem of 1975 solves this problem for an arbitrary regular family of curves on subdomains of the euclidean plane. The uniformization theorem also has an equivalent statement in terms of closed riemannian 2manifolds.
In mathematics, specifically in differential geometry, isothermal coordinates on a riemannian. Deturck, the cauchy problem for lorentz metrics with prescribed ricci curvature philippe g. From those, some other global quantities can be derived by. In a nutshell, our method is based on the following observation. The discovery that there is more than one geometry was of foundational significance and contradicted the german philosopher immanuel kant although some of the theorems of riemannian geometry are identical to those of euclidean, most differ. Riemannian connections, brackets, proof of the fundamental theorem of riemannian geometry, induced connection on riemannian submanifolds, reparameterizations and speed of geodesics, geodesics of the poincares upper half plane. Riemannian geometry is an expanded edition of a highly acclaimed and successful textbook originally published in portuguesefor firstyear graduate students in mathematics and physics. While more is said about this in a later section, it is worth emphasising that visual images, particularly those, which can be manipulated on the computer screen, invite students to observe and conjecture generalisations. If the riemannian manifold is oriented, some authors insist that a coordinate system must. Smoothness and regularity of solutions local differential geometry 53b20 local riemannian geometry elliptic equations and systems 35j60 nonlinear elliptic equations partial differential equations on manifolds. Deturck and kazdan, some regularity theorems in riemannian geometry ann. For instance, in the bonniemeyer theorem, we use the curvature of a space to learn about its metric. Regularity of solutions of the isoperimetric problem that. Harmonic mapping to generate harmonic coordinates in regions with boundary.
Some regularity theorems in riemannian geometry deturck, dennis m. Some integral geometry problems on finsler and riemannian. The starting point will be the boundary rigidity and conjugacy rigidity problems. On the other hand, there is no regularity theory of a. Comparison theorems in riemannian geometry jeff cheeger david g. Deturck, existence of metrics with prescribed ricci curvature. Riemannian geometry is an expanded edition of a highly acclaimed and successful textbook originally published in portuguese for firstyear graduate students in mathematics and physics. Some parabolic and elliptic problems in complex riemannian geometry by bin guo dissertation director. In particular, i do not treat the rauch comparison the orem, the morse index theorem, toponogovs theorem, or their important applications such as the sphere theorem, except to mention some of them. Eudml some regularity theorems in riemannian geometry. The lectures were to provide background for the analytic matters covered elsewhere during the conference and. Assylbekovintegral geometry problems on finsler and riemannian surfaces. The main themes here are the effect of the curvature on the usual notions of classical euclidean geometry, and the new notions and ideas motivated.
Differential geometry seminar, 19791980 to appear in annals of math. Next comes a detailed presentation of homogeneous spaces in which the main goal is to find formulas for their curvature. As an illustration, we describe some applications to the study of einstein metrics in dimension 4. They begin with a very concise introduction to riemannian geometry, followed by an exposition of toponogovs theorem the first such treatment in a book in english.
All known examples of length minimizing curves are smooth. Some regularity theorems in riemannian geometry numdam. The techniques used are, among other, the standards outils of linear elliptic analysis and comparison theorems of riemannian geometry, allards regularity theorem for minimizing varifolds, the isometric immersion theorem of nash and a parametric version due to gromov. In their theory one is concerned with the compactness of the class of riemannian manifolds with b ounded curv ature and lo w er b. The reciprocal ofthe fundamental theorem of riemannian. Differentiable control metrics and scaled bump functions. Anderson, convergence and rigidity of manifolds under ricci curvature bounds, invent. Chapter 2 studies sobolev spaces on riemannian manifolds. The reader is assumed to have some knowledge of standard manifold theory, including basic theory of tensors, forms, and lie groups. Successively, we will treat density problems, the sobolev imbedding theorem, the kondrakov theorem, and the study of the limiting case of the sobolev imbedding theorem. Pdf comparison theorems in riemannian geometry klaus. The regularity problem for sub riemannian geodesics roberto monti 1 introduction one of the main open problems in sub riemannian geometry is the regularity of length minimizing curves, see 12, problem 10.
Cheegergromov theory and applications to general relativity. When the ricci curvature is nonnegative, logconcave. We have comparison theorems, where we use the geometry of the space to get information about the metric. Compression bounds for lipschitz maps from the heisenberg group to l1. Its objectives are to deal with some basic problems in geometry and to provide a valuable tool for the researchers. Rigidity theorems in riemannian geometry christopher b. November 16, 2018 the fundamental theorem of riemannian geometry is inverted for analytic christo. Some regularity theorems for carnotcaratheodory metrics. Sprays, linear connections, riemannian manifolds, geodesics, canonical connection, sectional curvature and metric structure. Global riemannian geometry, including pinching see. Lefloch, injectivity radius and optimal regularity of lorentzian manifolds with bounded curvature.
What can be said about a complete riemannian manifold when mainly lower bounds for the sectional or ricci curvature are given. Burstall department of mathematical sciences university of bath introduction my mission was to describe the basics of riemannian geometry in just three hours of lectures, starting from scratch. Some regularity theorems in riemannian geometry eudml. A topological space xis second countable if xadmits a countable basis of open sets. Meanvalue theorems for riemannian manifolds volume 92 issue 34 a. Comparison theorems in riemannian geometry, northholland, amsterdam, 1975. An extension of rauchs metric comparison theorem and some applications, illinois j. Pdf comparison theorems for conjugate points in sub. Needle decompositions in riemannian geometry boaz klartag. Some exercises are included at the end of each section to give you something to think about. May 30, 2002 1 introduction the purpose of this chapter is to survey some recent results and state open questions concerning the rigidity of riemannian manifolds.
Structure theory and convergence in riemannian geometry. Notes for math 230a, differential geometry 7 remark 2. Terse notes on riemannian geometry tom fletcher january 26, 2010 these notes cover the basics of riemannian geometry, lie groups, and symmetric spaces. Comparison theorems for conjugate points in sub riemannian geometry.
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