Sasakian Geometry, Homotopy Spheres and Positive Ricci Curvature

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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 Σ²ⁿ⁺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 2²ᵐ distinct diffeomorphism types of homotopy real projective spaces in dimension 4m+1.