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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
\(\lim\sup_{z\in B_{0},z\rightarrow y}\) \(\overline{k}\left( z\right) <+\infty\);
\(\lim\sup_{z\in B_{0},z\rightarrow y}\) \(\overline{k}\left( z\right) d^{2}\left( y,z\right) =0\);
the action of \(K_{x}\) on \(H_{x}\) is polar;
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.

53C20 Global Riemannian geometry, including pinching
53C12 Foliations (differential geometric aspects)
57R30 Foliations in differential topology; geometric theory
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