Peter peterson's 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?

Peter peterson's book "Riemannnian Geometry" P351 says:Closed p351 says:

1. Closed manifold has no nontrivial totally convex subset. Using the energy functional if A⊂M $A\subset M$ is totally convex,then A⊂M convex, then $A\subset M$ is k-connected $k$-connected for any k,Why? It's however,not $k$.

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

Why are these statements true?

Peter peterson's book "Riemannnian Geometry" P351 says:Closed manifold has no nontrivial totally convex subset. Using the energy functional if A⊂M is totally convex,then A⊂M is k-connected for any k,Why? It's however,not possible for a closed n-manifold to have n-connected nontrivial subsets as this will violate Poincare duality,Why?