Nonnegatively curved quotient spaces with boundary.

*(English)*Zbl 1445.53028The 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.

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.

Reviewer: Vasyl Gorkaviy (Kharkov)

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

##### Keywords:

nonnegative curvature; isometric action; soul; torus manifold; rational elliptic manifold; convex set; distance function
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\textit{W. Spindeler}, Bol. Soc. Mat. Mex., III. Ser. 26, No. 2, 719--747 (2020; Zbl 1445.53028)

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##### References:

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