By Ballico E.

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38 (Cartan structure equations). 50) Rm ji = dωij − ωik ∧ ωkj . 51) Proof. 50) follows. 51) follows. For a surface M 2 , we have dω 1 = ω 2 ∧ ω21 , dω 2 = ω 1 ∧ ω12 , Rm 12 = dω21 . In particular, the Gauss curvature is given by K R (e1 , e2 ) e2 , e1 = Rm 12 (e1 , e2 ) = dω21 (e1 , e2 ) . 39 (Formula for connection 1-forms). Show that dω k (ei , ej ) = ωik (ej ) − ωjk (ei ) . 50) derive the formula for the connection 1-forms: 1 dω i (ej , ek ) + dω j (ei , ek ) − dω k (ej , ei ) . 4). 52) ωik (ej ) = 2.

57. 76). e. on M n . 57 can be given as follows. e. On the other hand it is easy 5For C is the radial graph of a continuous function. 5 for a discussion of Riemannian measure. 3. LAPLACIAN AND HESSIAN COMPARISON THEOREMS 31 to see that dp is not C 1 at those points x in Cut (p) for which there are two distinct minimal geodesics joining p to x. Thus, this set of points in Cut (p) has measure zero. We also know that by Sard’s theorem, the points in Cut (p) which are conjugate points form a measure zero set (since these points are singular values of expp ).

When M n is compact, the injectivity radius is always positive. 1. Laplacian comparison theorem. The idea of comparison theorems is to compare a geometric quantity on a Riemannian manifold with the corresponding quantity on a model space. Typically, in Riemannian geometry, model spaces have constant sectional curvature. As we shall see later, model spaces for Ricci flow are gradient Ricci solitons. 59 (Laplacian Comparison). 68) √ H coth √ Hdp (x) . 68) holds in the sense of distributions. That is, for any nonnegative C ∞ function ϕ on M n with compact support, we have √ √ Hdp ϕdµ.