my goals in this book on riemannian geometry are essentially the same as those which guided me in my eigenvalues in riemannian geometry [69], to introduce the subject, to coherently present a number of its basic techniques and results with a mind to future work, and to present some of the results that are attractive in their own right. this book differs from eigenvalues in that it starts at a more basic level,and therefore, it must present a broader view of the ideas from which all the various directions emerge. at the same time, other treatments of riemannian geometry are available at varying levels and interests,so i need not introduce everything. i have, therefore, attempted a viable introduction to riemannian geometry for a very broad group of students, with emphases and developments in areas not covered by other books.
目录
Preface 1 Riemannian manifolds 1.1 Connections 1.2 Parallel translation of vector fields 1.3 Geodesics and the exponential map 1.4 The torsion and curvature tensors 1.5 Riemannian metrics 1.6 The metric space structure 1.7 Geodesics and completeness 1.8 Calculations with moving frames 1.9 Notes and exercises 2 Riemannian curvature 2.1 The Riemann sectional curvature 2.2 Riemannian submanifolds 2.3 Spaces of constant sectional curvature 2.4 First and second variation of arc length 2.5 Jacobi''s equation and criteria 2.6 Elementary comparison theorems 2.7 Jacobi fields and the exponential map 2.8 Riemann normal coordinates 2.9 Notes and exercises 3 Riemannian volume 3.1 Geodesic spherical coordinates 3.2 The conjugate and cut loci 3.3 Riemannian measure 3.4 Volume comparison theorems 3.5 The area of spheres 3.6 Fermi coordinates 3.7 Integration of differential forms 3.8 Notes and exercises 3.9 Appendix: Eigenvalue comparison theorems 4 Riemannian coverings 4.1 Riemannian coverings 4.2 The fundamental group 4.3 Volume growth Of Riemannian coverings 4.4 Discretization of Riemannian manifolds 4.5 The free homotopy classes 4.6 Gauss-Bonnet theory of surfaces 4.7 Notes and exercises 5 The kinematic density 5.1 The differential geometry of analytical dynamics 5.2 Santalo''s formula 5.3 The Berger-Kazdan inequalities 5.4 On manifolds with no conjugate points 5.5 Notes and exercises 6 Isoperimetric inequalities 6.1 Isoperimetric constants 6.2 The isoperimetric inequality in Euclidean space 6.3 The isoperimetric inequality on spheres 6.4 Symmetrization and isoperimetric inequalities 6.5 Buser''s isoperimetric inequality 6.6 Croke''s isoperimetric inequality 6.7 Discretizations and isoperimetry 6.8 Notes and exercises 7 Comparison and finiteness theorems 7.1 Preliminaries 7.2 H.E. Rauch''s comparison theorem 7.3 Comparison theorems with initial submanifolds 7.4 Refinements of the Rauch theorem 7.5 Triangle comparison theorems 7.6 Convexity 7.7 Center of mass 7.8 Cheeger''s finiteness theorem 7.9 Notes and exercises Hints and sketches of solutions Bibliography Index
