Curvature, diameter, and quotient manifolds

Details Curvature, diameter, and quotient manifolds Curvature, diameter, and quotient manifolds

2002-09-13

arxiv.org/abs/math/0209173v1

Mathematics

Differential Geometry

12 pages

Scientific paper

Gromov showed that there is an upper bound on the Betti numbers of all closed Riemannian n-manifolds of nonnegative sectional curvature. Grove asked whether such manifolds (if simply connected) fall into only finitely many rational homotopy types. We give a negative answer, in fact in dimension 6, which is the smallest possible. We also give counterexamples to some related questions in dimensions 7 and 9, improving the original counterexamples by Fang and Rong which were in dimensions at least 22.

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