Peter Peterson's book "Riemannnian Geometry" p351 says:
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 $k$.
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?