Lytchak, Alexander; Thorbergsson, Gudlaugur Curvature explosion in quotients and applications. (English) Zbl 1221.53067 J. Differ. Geom. 85, No. 1, 117-140 (2010). 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. Reviewer: I. G. Nikolaev (Urbana) Cited in 28 Documents 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 × Cite Format Result Cite Review PDF Full Text: DOI arXiv Euclid