MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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?

share|cite|improve this question
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.

share|cite|improve this answer
thank you,but in the book the soul theorem is after the second can you give me another explaination about poincare dual? – jiangsaiyin Sep 2 '12 at 12:20
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.