This volume introduces techniques and theorems of Riemannian geometry, and opens the way to advanced topics. The text combines the geometric parts of Riemannian geometry with analytic aspects of the theory, and reviews recent research. The updated second edition includes a new coordinate-free formula that is easily remembered (the Koszul formula in disguise); an expanded number of coordinate calculations of connection and curvature; general fomulas for curvature on Lie Groups and submersions; variational calculus integrated into the text, allowing for an early treatment of the Sphere theorem using a forgotten proof by Berger; recent results regarding manifolds with positive curvature.

It suffices to check the first two cases as the third is similar to the second. For the first we can assume that z = x k and find l such that Hence, For the second case select l, m with The assumption about the metrics on A and B then lead to Example 11.1.6. Suppose with the usual metric induced from Then we have a Riemannian submersion whose fibers have diameter as Using the previous example it follows that in the Gromov-Hausdorff topology. Example 11.1.7. One can similarly see that

coordinates is of that form Thus The above theorem has an almost immediate corollary. Corollary 11.2.3. If in addition we assume that , , then there is a constant such that And on a domain with smooth boundary, The Schauder estimates can be used to show that the Dirichlet problem always has a unique solution. Theorem 11.2.4. Suppose is a bounded domain with smooth boundary. Then the Dirichlet problem always has a unique solution if and Observe that uniqueness is an immediate

superlevel set is noncompact. If a > 0, then is also noncompact. So we can assume that a ≤ 0. As all of the Busemann functions b c are zero at p also In particular, p ∈ A. Using noncompactness select a sequence p k ∈ A that goes to infinity. Then consider segments , and as in the construction of rays, choose a subsequence so that converges. This forces the segments to converge to a ray emanating from p. As A is totally convex, all of these segments lie in A. Since A is closed the ray must also

vector field along c, then we can compute and then project to obtain the derivative of V along c in M. Example 6.1.1 shows what can go wrong if we are not careful about projecting the derivatives. 6.1.2 Third Partials One of the uses of taking derivatives of vector fields along curves is that we can now define third and higher order partial derivatives. If we wish to compute then consider the vector field and define Something rather interesting happens with this definition. We expected and

form In case the vector fields come from a proper variation of c this is equal to the second variation of energy. Assume below that locally minimizes the energy functional. This implies that for all proper variations. (1)If for a proper variation, then V is a Jacobi field. Hint: Let W be any other variational field that also vanishes at the end points and use that for all small to show that . Then use that this holds for all W to show that V is a Jacobi field. (2)Let V and J be variational