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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?

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up vote 8 down vote accepted

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.

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thank you,but in the book the soul theorem is after the second statement.so can you give me another explaination about poincare dual? –  jiangsaiyin Sep 2 '12 at 12:20
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Petersen's book is not meant to be read in a linear order. That every closed totally convex subset is a manifold is a basic fact, which does not require the soul theorem. –  Igor Belegradek Sep 2 '12 at 12:58
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