Could you give examples of totally geodesic subset of codim>1 in positively curved Alexandrov space?
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6$\begingroup$ 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 $\endgroup$– Benoît KloecknerCommented Sep 12, 2012 at 14:55
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2 Answers
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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 $\mathbb CP^n$.
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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, 2012 at 8:34
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$\begingroup$ In fact,I need Alexandrov space $\endgroup$ Commented Sep 12, 2012 at 8:49
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$\begingroup$ Round spheres are Alexandrov spaces. $\endgroup$– J. GECommented Sep 12, 2012 at 9:47
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$\begingroup$ I need Alexandrov space which is not manifold $\endgroup$ Commented Sep 12, 2012 at 12:03
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2$\begingroup$ Take any your favorite example of Riemannian manifold, then the spherical suspension over the manifold will do the job. $\endgroup$– J. GECommented Sep 12, 2012 at 12:49