Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature

Details Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature

2002-01-16

arxiv.org/abs/math/0201147v2

Topology 42 (2003), 981-1002.

Mathematics

Differential Geometry

22 pages, revised version with some clarifications and added references

Scientific paper

We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres $\scriptstyle{\Sigma^{2n+1}}$ the moduli space of Sasakian structures has infinitely many positive components determined by inequivalent underlying contact structures. We also prove the existence of Sasakian metrics with positive Ricci curvature on each of the known $\scriptstyle{2^{2m}}$ distinct diffeomorphism types of homotopy real projective spaces in dimension $4m+1$.

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