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Curvature explosion in quotients and applications. (English) Zbl 1221.53067
Let $$\left\langle M,g\right\rangle$$ be a Riemannian manifold and $$K$$ a closed group of isometries of $$\left\langle M,g\right\rangle$$. The authors study the quotient space $$B=M/K$$ (which is a possibly singular Aleksandrov space of curvature locally bounded below). Let $$B_{0}$$ denote the maximal stratum of $$B$$. Then, as $$z$$ approaches a singular point $$y\in B\backslash B_{0}$$, the curvature of $$B$$ may “explode”. For $$z\in B_{0}$$, let $$\overline{k}\left( z\right)$$ denote the maximum of the sectional curvatures at $$z$$. Let $$x\in M$$ be a point with the isotropy group $$K_{x}$$ acting on the normal space $$H_{x}$$ of the orbit $$Kx\subset M$$. Set $$y=Kx\in B$$. The authors main result describes points of “non-explosions”. The following conditions are equivalent. 8mm
(i)
$$\lim\sup_{z\in B_{0},z\rightarrow y}$$ $$\overline{k}\left( z\right) <+\infty$$;
(ii)
$$\lim\sup_{z\in B_{0},z\rightarrow y}$$ $$\overline{k}\left( z\right) d^{2}\left( y,z\right) =0$$;
(iii)
the action of $$K_{x}$$ on $$H_{x}$$ is polar;
(iv)
a neighborhood of $$y\in B$$ is a smooth Riemannian orbifold.
Applications of this result and generalizations to singular Riemannian foliations without horizontal conjugate points are given.

##### MSC:
 53C20 Global Riemannian geometry, including pinching 53C12 Foliations (differential geometric aspects) 57R30 Foliations in differential topology; geometric theory
##### Keywords:
Riemannian manifold; curvature; group of isometries; foliation
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