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?



Closed manifold has no nontrivial totally convex subset?
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"](http://www.intlpress.com/JDG/archive/1981/16-2-333.pdf) 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.


