Positively curved manifolds with maximal discrete symmetry rank

Details Positively curved manifolds with maximal discrete symmetry rank Positively curved manifolds with maximal discrete symmetry rank

2003-08-14

arxiv.org/abs/math/0308139v1

Mathematics

Differential Geometry

18 pages

Scientific paper

Let M be a closed simply connected n-manifold of positive sectional curvature. We determine its homeomorphism or homotopic type if M also admits an isometric elementary p-group action of large rank. Our main results are: There exists a constant p(n)>0 such that (1) If M^{2n} admits an effective isometric \Bbb Z_p^k-action for a prime p\ge p(n), then k\le n and ``='' implies that M^{2n} is homeomorphic to a sphere or a complex projective space. (2) If M^{2n+1} admits an isometric S^1 x \Bbb Z_p^k-action for a prime p\ge p(n), then k\le n and ``='' implies that M is homeomorphic to a sphere. (3) For M in (1) or (2), if n\ge 7 and k\ge [\frac{3n}4]+2, then M is homeomorphic to a sphere or homotopic to a complex projective space.

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