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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
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