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**A splitting theorem for nonnegatively curved Alexandrov spaces.**
*(English)*
Zbl 1261.53044

Let \(M\) be an \(n\)-dimensional compact Alexandrov space of nonnegative curvature. A “boundary stratum” of \(M\) is an extremal set of locally constant codimension \(1\). A “stratification” of \(\partial M\) is a cover of \(\partial M\) by boundary strata whose pairwise intersections have codimension at least \(2\).

The main result of the paper (Theorem 1.2) infers the metric product structure of \(M\) from the existence of a particular stratification of \(\partial M\). Namely, assume that \(F_1,\dots, F_{\ell+1}\) is a stratification for which there exists \(k\in\{1,\dots,\ell\}\) such that (i) the intersection \(F_1\cap\dots\cap F_{k+1}\) is empty; (ii) if any of the sets \(F_1,\dots,F_{k+1}\) is omitted from (i), the resulting intersection is nonempty. Then the conclusion is that \(M\) is isometric to the product \(S\times D\) where both factors are nonnegatively curved Alexandrov spaces, \(\dim S=n-k\), and \(\dim D=k\). Here, \(S\) can be chosen to be any of the intersections mentioned in (ii) above. Moreover, in the special case \(k=\ell\) the set \(S\) is a soul of \(M\).

Among several other corollaries, the author obtains the following: If \(M\) has at least \(n+2\) boundary strata, then it is a nontrivial metric product. The proofs employ a diverse array of methods of metric geometry and are carefully presented with precise pointers to the literature. Additional details are available in the author’s thesis on which this article is based.

The main result of the paper (Theorem 1.2) infers the metric product structure of \(M\) from the existence of a particular stratification of \(\partial M\). Namely, assume that \(F_1,\dots, F_{\ell+1}\) is a stratification for which there exists \(k\in\{1,\dots,\ell\}\) such that (i) the intersection \(F_1\cap\dots\cap F_{k+1}\) is empty; (ii) if any of the sets \(F_1,\dots,F_{k+1}\) is omitted from (i), the resulting intersection is nonempty. Then the conclusion is that \(M\) is isometric to the product \(S\times D\) where both factors are nonnegatively curved Alexandrov spaces, \(\dim S=n-k\), and \(\dim D=k\). Here, \(S\) can be chosen to be any of the intersections mentioned in (ii) above. Moreover, in the special case \(k=\ell\) the set \(S\) is a soul of \(M\).

Among several other corollaries, the author obtains the following: If \(M\) has at least \(n+2\) boundary strata, then it is a nontrivial metric product. The proofs employ a diverse array of methods of metric geometry and are carefully presented with precise pointers to the literature. Additional details are available in the author’s thesis on which this article is based.

Reviewer: Leonid Kovalev (Syracuse)

### MSC:

53C23 | Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces |

51H25 | Geometries with differentiable structure |

### Keywords:

Alexandrov space; nonnegative curvature; metric splitting; extremal subset; soul; boundary strata
Full Text:
DOI

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