Question

Peter Peterson's book "Riemannnian Geometry" p351 says:

1. Closed manifold has no nontrivial totally convex subset. Using the energy functional if $A\subset M$ is totally convex, then $A\subset M$ is $k$-connected for any $k$.

2. It is however not possible for a closed n-manifold to have $n$-connected nontrivial subsets as this will violate Poincare duality.

Why are these statements true?

Answer 1

The inclusion from a closed totally convex subset to the ambient manifold is a homotopy equivalence. Details can be found in "Totally convex sets in complete Riemannian manifolds" by Bangert, JDG, 1981.

For the second assertion, Cheeger-Gromoll prove in their paper on the soul theorem that any closed totally convex subset is a manifold with boundary, so if the boundary is non-empty, the manifold has zero top-dimensional homology, hence it cannot be homotopy equivalent to a closed manifold of that top dimension.