Combining an accessible style with precision analysis, the author introduces the main topics of Riemannian geometry. The results are stated without detailed proofs, enabling the reader to quickly obtain an overview of the entire field. However, since a Riemannian manifold is a subtle object, appealing to highly non-natural concepts, the first three chapters are devoted to introducing the various concepts and tools of Riemannian geometry in the most natural and motivating way following in particular Gauss and Riemann.
0. Vector fields, tensors
1. Tensor Riemannian duality, the connection and the curvature
2. The parallel transport
3. Absolute (Ricci) calculus, commutation formulas
4. Hodge and the Laplacian, Bochners technique
5. Generalizing Gauss-Bonnet, characteristic classes and C. GEOMETRIC MEASURE THEORY AND PSEUDO-HOLOMORPHIC B. HIGHER DIMENSIONS A. THE CASE OF SURFACES IN R3 C. various other bundles
3. Harmonic maps between Riemannian manifolds
4. Low dimensional Riemannian geometry
5. Some generalizations of Riemannian geometry
6. Gromov mm-spaces
7. Submanifolds B. Spinors A. Exterior differential forms (and some others) C. RICCI FLAT KAHLER AND HYPERKAHLER MANIFOLDS
6. Ka''hlerian manifolds (Ka''hler metrics)
Chapter XI SOME OTHER IMPORTANT TOPICS
1. Non compact manifolds
2. Bundles over Riemannian manifolds B. QUATERNIONIC-KAHLER MANIFOLDS A. G2 AND Spin(7) HIERRACHY HOLONOMY GROUPS AND KAHLER MANIFOLDS
1. Definitions and philosophy
2. Examples
3. General structure theorems
4. The classification result
5. The rare cases b. on a given compact manifolds closures
Chapter X GLOBAL PARALLEL TRANSPORT AND ANOTHER RIEMANNIAN a. collapsing C. THE CASE OF RICCI CURVATURE
12. Compactness, convergence results
13. The set of all Riemannian structures collapsing B. MORE FINITENESS THEOREMS A. CHEEGERs FINITENESS THEOREM
11. Finiteness results of all Riemannian structures third part Finiteness, compactness, collapsing and the space D. NEGATIVE VERSUS NONPOSITIVE CURVATURE
10. The negative side Ricci curvature C. VOLUMES, FUNDAMENTAL GROUP B. QUASI-ISOMETRIES A. INTRODUCTION
E. POSSIBLE APPROACHES, LOOKING FOR THE FUTURE
7. Ricci curvature positive, nonnegative and just below
8. The positive side scalar curvature
9. The negative side sectional curvature D. POSITIVITY OF THE CURVATURE OPERATOR C. THE NON-COMPACT CASE B. HOMOLOGY TYPE AND THE FUNDAMENTAL GROUP A. THE KNOWN EXAMPLES
6. The positive side sectional curvature second part Curvature of a given sign
1. Introduction
2. The positive pinching
3. Pinching around zero
4. The negative pinching
5. Ricci curvature pinching first part Pinching problems b. hierarchy of curvaturesa. hopfs urge d. the set of constants, ricci flat metrics
18. The Yamabe problem
Chapter IX from curvature to topology
0. Some history and structure of the
chapter c. moduli b. uniqueness a. existence b. homogeneous spaces and others
14. Examples from Analysis I the evolution Ricci flow
15. Examples from Analysis II the Ka''hler case
16. The sporadic examples
17. Around existence and uniqueness a. symmetric spaces THIRD PART EINSTEIN MANIFOLDS
12. Hilberts variational principle and great hopes
13. The examples from the geometric hierachy
10. The case of Min R d/2 when d=4
11. Summing up questions on MinVol and Min(R) d/2 b. the simplicial volume of gromov a. using integral formulas d. cheeger-rong examples
9. Some cases where MinVol > 0 , Min Rd/2 > 0 c. nilmanifolds and the converse almost flat manifolds b. wallachs type examples a. s1 fibrations and more examples MinDiam = 0 MinVol, MinDiam
5. Definitions
6. The case of surfaces
7. Generalities, compactness, finiteness and equivalence
8. Cases where MinVol = Min R d/2 = 0 and SECOND PART WHICH METRIC IS THE LESS CURVED Min R d/2 , FIRST PART: PURE GEOMETRIC FUNCTIONALS
1. Systolic quotients
2. Counting periodic geodesics
3. The embolic volume
4. Diameter/Injectivity riemannian metric on a given compact manifold ?
0. Introduction
and a possible scheme of attack c. the structure on a given Sd and KPn
19. Inverse problems II conjugacy of geodesics flows
Chapter VIII the search for distinguished metrics what is the best b. bott and samelson theorems a. definitions and the need to be careful are closed
14. The case of negative curvature
15. The case of nonpositive curvature
16. Entropies on various space forms
17. From Osserman to Lohkamp
18. Inverse problems I manifolds all of whose geodesics b. the various notions of entropya. er
