Gallot, S: Riemannian Geometry

fields along parameterized submanifolds.- First variation formula.- Second variation formula.- C. Jacobi Vector Fields.- Basic topics about second derivatives.- Index form.- Jacobi fields and exponential map.- Applications: Sn, Hn, PnR, 2-dimensional manifolds.- D. Riemannian Submersions and Curvature.- Riemannian submersions and connexions.- Jacobi fields of PnC.- O'Neill's formula.- Curvature and length of small circles. Application to Riemannian submersions.- E. The Behavior of Length and Energy in the Neighborhood of a Geodesic.- The Gauss lemma.- Conjugate points.- Some properties of the cut-locus.- F. Manifolds with Constant Sectional Curvature.- Spheres, Euclidean and hyperbolic spaces.- G. Topology and Curvature.- The Myers and Cartan theorems.- H. Curvature and Volume.- Densities on a differential manifold.- Canonical measure of a Riemannian manifold.- Examples: spheres, hyperbolic spaces, complex projective spaces.- Small balls and scalar curvature.- Volume estimates.- I. Curvature and Growth of the Fundamental Group.- Growth of finite type groups.- Growth of the fundamental group of compact manifolds with negative curvature.- J. Curvature and Topology.- Traditional point of view: pinched manifolds.- Almost flat pinching.- Coarse point of view: compactness theorems of Gromov and Cheeger.- K. Curvature and Representations of the Orthogonal Group.- Decomposition of the space of curvature tensors.- Conformally flat manifolds.- The second Bianchi identity.- Chapitre IV: Analysis on Manifolds and the Ricci Curvature.- A. Manifolds with Boundary.- Definition.- The Stokes theorem and integration by parts.- B. Bishop's Inequality Revisited.- Some commutations formulas.- Laplacian of the distance function.- Another proof of Bishop's inequality.- The Heintze-Karcher inequality.- C. Differential forms and Cohomology.- The de Rham complex.- Differential operators and their formal adjoints.- The Hodge-de Rham theorem.- A second visit to the Bochner method.- D. Basic Spectral Geometry.- The Laplace operator and the wave equation.- Statement of the basic results on the spectrum.- E. Some Examples of Spectra.- The spectrum of flat tori.- Spectrum of (Sn, can).- F. The Minimax Principle.- The basic statements.- V. Riemannian Submanifolds.