×

zbMATH — the first resource for mathematics

Nonnegatively curved quotient spaces with boundary. (English) Zbl 1445.53028
The paper under review concerns topological properties of nonnegatively curved Riemannian manifolds. Particularly, Riemannian manifolds admitting certain symmetries are considered, c.f. [K. Grove and C. Searle, J. Pure Appl. Algebra 91, No. 1–3, 137–142 (1994; Zbl 0793.53040); J. Differ. Geom. 47, No. 3, 530–559 (1997; Zbl 0929.53017); B. Wilking, Ann. Math. 163, 607–668 (2006; Zbl 1104.53030)].
As the main result, the following statement is proved.
Theorem. Let \((M,g)\) be a compact, connected and nonnegatively curved Riemannian manifold admitting an isometric action by a compact Lie group \(G\) so that the quotient space \(M/ G\) has non-empty boundary. Let \(\pi: M \to M/ G\) denote the quotient map and \(B\) be an arbitrary boundary stratum of \(M/ G\). Then there exists a closed smooth \(G\)-invariant submanifold \(N\subset M\) such that \(M\backslash \pi^{-1}(B)\) is equivariantly diffeomorphic to the normal bundle of \(N\).
The proof is based on a careful analysis of geometrical and analitycal properties of convex subsets and distance functions on \(M/ G\) followed by specific soul constructions for manifolds in question.
The proved theorem is claimed to be useful for classification of nonnegatively curved Riemannian manifolds with symmetries. For instance, the following statement related to the Bott conjecture on nonnegatively curved Riemannian manifolds is obtained.
Theorem. Let \(M\) be a compact and simply connected torus manifold admitting an invariant metric of nonnegative curvature. Then \(M\) is rationally elliptic.
MSC:
53C20 Global Riemannian geometry, including pinching
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Cheeger, J.; Gromoll, D., On the structure of complete manifolds of nonnegative curvature, Ann. Math., 2, 96, 413-443 (1972) · Zbl 0246.53049
[2] Galaz-Garcia, F., Nonnegatively curved fixed point homogeneous manifolds in low dimensions, Geom. Dedicat., 157, 367-396 (2012) · Zbl 1254.53059
[3] Galaz-Garcia, F.; Spindeler, W., Nonnegatively curved fixed point homogeneous 5-manifolds, Ann. Glob. Anal. Geom., 41, 2, 253-263 (2012) · Zbl 1239.53046
[4] Grove, K.; Halperin, S., Dupin hypersurfaces, group actions and the double mapping cylinder, J. Differ. Geom., 26, 3, 429-459 (1987) · Zbl 0637.53075
[5] Grove, K.; Halperin, S., Contributions of rational homotopy theory to globalproblems in geometry, Publ. Math. IHES, 56, 1, 171-177 (1982) · Zbl 0508.55013
[6] Grove, K.: Geometry of, and via, symmetries. In: Conformal, Riemannian and Lagrangian geometry (Knoxville, TN, 2000), volume 27 of Univ. Lecture Ser., American Mathematical Society, Providence, RI, pp. 31-53 (2002)
[7] Grove, K.; Searle, C., Positively curved manifolds with maximal symmetry-rank, J. Pure Appl. Algebra, 91, 1-3, 137-142 (1994) · Zbl 0793.53040
[8] Grove, K.; Searle, C., Differential topological restrictions curvature and symmetry, J. Differ. Geom., 47, 3, 530-559 (1997) · Zbl 0929.53017
[9] Perelman, G.: Alexandrov’s spaces with curvatures bounded from below II. (1991). https://anton-petrunin.github.io/papers/alexandrov/perelmanASWCBFB2+.pdf
[10] Petrunin, A., Semiconcave functions in Alexandrov’s geometry, Surveys Differ Geom, 11, 1, 137-202 (2006) · Zbl 1166.53001
[11] Spindeler, W.: \({\sf S}^1\)-actions on \(4\)-manifolds and fixed point homogeneous manifolds of nonnegative curvature. Dissertation, WWU Münster (2014). http://miami.uni-muenster.de/Record/272b3efb-9d8d-4ee3-8b15-2ab860f49ed0 · Zbl 1297.53006
[12] Spindeler, W.: \({\sf S}^1\)-actions on 4-manifolds and fixed point homogeneous manifolds of nonnegative curvature (2015). arXiv:1510.01548
[13] Wörner, A.: Boundary Strata of non-negatively curved Alexandrov spaces and a splitting theorem. Dissertation, WWU Münster (2010) · Zbl 1194.53002
[14] Wiemeler, M., Torus manifolds and non-negative curvature, J. Lond. Math. Soc. II. Ser., 91, 3, 667-692 (2015) · Zbl 1321.57043
[15] Wilking, B., Positively curved manifolds with symmetry, Ann. Math., 163, 607-668 (2006) · Zbl 1104.53030
[16] Wilking, B., A duality theorem for riemannian foliations in nonnegative sectional curvature, Geom. Funct. Anal., 17, 4, 1297-1320 (2007) · Zbl 1139.53014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.