## 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.

### MSC:

 53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces 51H25 Geometries with differentiable structure
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### References:

 [1] S B Alexander, R L Bishop, A cone splitting theorem for Alexandrov spaces, Pacific J. Math. 218 (2005) 1 · Zbl 1125.53023 [2] D Burago, Y Burago, S Ivanov, A course in metric geometry, Graduate Studies in Mathematics 33, American Mathematical Society (2001) · Zbl 0981.51016 [3] Y Burago, M Gromov, G Perelman, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3, 222 · Zbl 0802.53018 [4] J Cheeger, D Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72) 119 · Zbl 0223.53033 [5] J Cheeger, D Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. 96 (1972) 413 · Zbl 0246.53049 [6] K Grove, P Petersen, A radius sphere theorem, Invent. Math. 112 (1993) 577 · Zbl 0801.53029 [7] V Kapovitch, Perelman’s stability theorem, Surv. Differ. Geom. 11, Int. Press, Somerville, MA (2007) 103 · Zbl 1151.53038 [8] A Lytchak, Allgemeine Theorie der Submetrien und verwandte mathematische Probleme, Bonner Mathematische Schriften 347, Universität Bonn Mathematisches Institut (2002) · Zbl 1020.54021 [9] A Lytchak, Open map theorem for metric spaces, Algebra i Analiz 17 (2005) 139 · Zbl 1152.53033 [10] Y Mashiko, A splitting theorem for Alexandrov spaces, Pacific J. Math. 204 (2002) 445 · Zbl 1058.53036 [11] S J Mendon\cca, The asymptotic behavior of the set of rays, Comment. Math. Helv. 72 (1997) 331 · Zbl 0910.53026 [12] A D Milka, Metric structure of a certain class of spaces that contain straight lines, Ukrain. Geometr. Sb. Vyp. 4 (1967) 43 [13] A Mitsuishi, A splitting theorem for infinite dimensional Alexandrov spaces with nonnegative curvature and its applications, Geom. Dedicata 144 (2010) 101 · Zbl 1191.53051 [14] Y Otsu, Differential geometric aspects of Alexandrov spaces, Math. Sci. Res. Inst. Publ. 30, Cambridge Univ. Press (1997) 135 · Zbl 0891.53025 [15] Y Otsu, T Shioya, The Riemannian structure of Alexandrov spaces, J. Differential Geom. 39 (1994) 629 · Zbl 0808.53061 [16] G Perelman, Alexandrov’s spaces with curvatures bounded from below II, preprint (1991) [17] G Perelman, Elements of Morse theory on Aleksandrov spaces, Algebra i Analiz 5 (1993) 232 · Zbl 0815.53072 [18] G Perelman, Proof of the soul conjecture of Cheeger and Gromoll, J. Differential Geom. 40 (1994) 209 · Zbl 0818.53056 [19] G Perelman, Collapsing with no proper extremal subsets, Math. Sci. Res. Inst. Publ. 30, Cambridge Univ. Press (1997) 149 · Zbl 0887.53049 [20] G Perelman, A M Petrunin, Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, Algebra i Analiz 5 (1993) 242 · Zbl 0802.53019 [21] G Perelman, A Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, preprint (1995) [22] A Petrunin, Applications of quasigeodesics and gradient curves, Math. Sci. Res. Inst. Publ. 30, Cambridge Univ. Press (1997) 203 · Zbl 0892.53026 [23] A Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8 (1998) 123 · Zbl 0903.53045 [24] A Petrunin, Semiconcave functions in Alexandrov’s geometry, Surv. Differ. Geom. 11, Int. Press, Somerville, MA (2007) 137 · Zbl 1166.53001 [25] V A Sharafutdinov, Convex sets in a manifold of nonnegative curvature, Mat. Zametki 26 (1979) 129, 159 · Zbl 0439.53052 [26] T Shioya, Splitting theorems for non-negatively curved open manifolds with large ideal boundary, Math. Z. 212 (1993) 223 · Zbl 0791.53045 [27] M Strake, A splitting theorem for open nonnegatively curved manifolds, Manuscripta Math. 61 (1988) 315 · Zbl 0653.53023 [28] V A Toponogov, Riemannian spaces which contain straight lines, Amer. Math. Soc. Transl. Ser. 2 37 (1964) 287 · Zbl 0138.42902 [29] G Walschap, A splitting theorem for 4-dimensional manifolds of nonnegative curvature, Proc. Amer. Math. Soc. 104 (1988) 265 · Zbl 0662.53033 [30] B Wilking, Positively curved manifolds with symmetry, Ann. of Math. 163 (2006) 607 · Zbl 1104.53030 [31] A Wörner, Boundary strata of nonnegatively curved Alexandrov spaces and a splitting theorem, PhD thesis, WWU Münster (2010) · Zbl 1194.53002 [32] J W Yim, Space of souls in a complete open manifold of nonnegative curvature, J. Differential Geom. 32 (1990) 429 · Zbl 0733.53017
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