Curvature, Diameter and Betti Umbers
نویسنده
چکیده
Let V denote a compact (without boundary) connected Riemannian manifold of dimension n. We denote by K the sectional curvature of V and we set inf K = ird~ K(-r) where 9 runs over all tangent 2-planes in V. One calls V a manifold of non-negative curvature ff inf K>~0. This condition has the following geometric meaning. A n n-dimensional Riemannian manifold has non-negative curvature iff for each point v ~ V there is a positive number e and a map f of the n-dimensional Euclidean e-ball B into V with the following two properties: (a) f sends B diffeomorphicly onto the e-ball in V with the center v. (b) The map f is distance non-increasing, that is for any two points x and y in B one has.
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