basicelements of differential geometry and topology

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Cheeger-Gromoll splitting theorem and manifolds of nonnegative curvature. In the study of manifolds with nonnegative curvature, often (especially when the curvature is not strictly positive) the manifolds split as the product of a lower dimensional manifold with a line. Recall that a geodesic line is a unit speed geodesic γ : (−∞, ∞) → M n such that the distance between any points on γ is the length of the arc of γ between those two points; that is, for any s1 , s2 ∈ (−∞, ∞) , d (γ (s1 ) , γ (s2 )) = |s2 − s1 | .

The injectivity radius of a Riemannian manifold is defined to be inj (M n , g) inf {inj (p) : p ∈ M n } . 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.

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.

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