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Could you give examples of totally geodesic subset of codim>1 in positively curved Alexandrov space?

asked Sep 12 at 8:29

5

Given your comment below, it is clear to me that your question lacks motivation and context. What do you need exactly, and why? Whithout these precisions, you will not get the answers you expect. Read the FAQ about how to ask a question – Benoît Kloeckner Sep 12 at 14:55

J. GE · Answer 1 · 2012-09-12 12:50:30Z UTC

For the Fubini–Study metric on $\mathbb CP^n$ (it has positive 1/4-pinched curvature). It has totally geodesic embedded $\mathbb CP^{n-1}$ has codimension $2$. Then you can construct more larger codimension ones. And more trivial examples are given by closed geodesics in any positively curved space.

Edit: If you prefer non-Riemannian Alexandrov space, take the spherical suspension of $\mathbb CP^n$.

Benoît Kloeckner · Answer 2 · 2012-09-12 08:34:26Z UTC

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The obvious answer is an equatorial sphere (= intersection with a linear subspace of any codimension) in the unit sphere of $\mathbb{R}^n$.

Without more details on your motivation, it is difficult to judge whether this answer is satisfying or not.

answered Sep 12 at 8:34

Benoît Kloeckner
5,741●18●30

In fact,I need Alexandrov space – jiangsaiyin Sep 12 at 8:49

Round spheres are Alexandrov spaces. – J. GE Sep 12 at 9:47

I need Alexandrov space which is not manifold – jiangsaiyin Sep 12 at 12:03

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Take any your favorite example of Riemannian manifold, then the spherical suspension over the manifold will do the job. – J. GE Sep 12 at 12:49

examples of totally geodesic subset

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