$i$-convexity of manifolds with real projective structures

Abstract:

We compare the notion of higher-dimensional convexity, as defined by Carrière, for real projective manifolds with the existence of hemispheres. We show that if an i-convex real projective manifold M of dimension n for an integer i with $0 < i < n$ has an i-dimensional hemisphere, then M is projectively homeomorphic to ${{\mathbf {S}}^n}/\Gamma$ where $\Gamma$ is a finite subgroup of $O(n + 1,{\mathbf {R}})$ acting freely on ${{\mathbf {S}}^n}$.