Abstract
In this paper we prove that an isometry between orbit spaces of two proper isometric actions is smooth if it preserves the codimension of the orbits or if the orbit spaces have no boundary. In other words, we generalize Myers–Steenrod’s theorem for orbit spaces. These results are proved in the more general context of singular Riemannian foliations.
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Alexandrino, M.M., Lytchak, A.: On smoothness of isometries between orbit spaces, Riemannian geometry and applications. In: Proceedings RIGA, pp. 17–28. Ed. Univ. Bucureşti (2011)
Alexandrino, M.M., Töben, D.: Equifocality of singular Riemannian foliations. Proc. Am. Math. Soc. 136(9), 3271–3280 (2008)
Chevalley, C.: Invariants of nite groups generated by reections. Am. J. Math. 77, 778–782 (1955)
Evans, L.C.: Partial Differential Equations, Graduate Studies in Mathematics 19. American Mathematical Society, Providence (1998)
Ferus, D., Karcher, H., Münzner, H.F.: Cliffordalgebren und neue isoparametrische Hyperflächen. Math. Z. 177(4), 479–502 (1981)
Gromoll, D., Walschap, G.: Metric Foliations and Curvatures, Progress in Mathematics 268. Birkhäuser, Basel (2009)
Lytchak, A., Thorbergsson, G.: Curvature explosion in quotients and applications. J. Differ. Geom. 85, 117–139 (2010)
Luna, D., Richardson, R.W.: A generalization of the Chevalley restriction theorem. Duke Math. J. 46, 487–496 (1979)
Molino, P.: Riemannian Foliations, Progress in Mathematics 73. Birkhäuser, Boston (1988)
Palais, R.S., Terng, C.-L.: Critical Point Theory and Submanifold Geometry. Lecture Notes in Mathematics. Springer, Berlin (1988)
Schwarz, G.W.: Lifting smooth homotopies of orbit spaces. Publ. Math. L’I.H.É.S. 51, 37–135 (1980)
Strub, R.: Local classification of quotients of smooth manifolds by discontinuous groups. Math. Z. 179, 43–57 (1982)
Swartz, E.: Matroids and quotients of spheres. Math. Z. 241(2), 247–269 (2002)
Acknowledgments
The authors are grateful to Alexander Lytchak for inspiring the main questions of this work, and for very helpful discussions and suggestions. The authors also thank Wolfgang Ziller, Dirk Töben, Ricardo Mendes, Renato Bettiol and the referee for useful suggestions.
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The first author was supported by a research productivity scholarship from CNPq-Brazil and partially supported by FAPESP (São Paulo, Brazil). The second author was partially supported by Benjamin Franklin Fellowship at the University of Pennsylvania.
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Alexandrino, M.M., Radeschi, M. Isometries between leaf spaces. Geom Dedicata 174, 193–201 (2015). https://doi.org/10.1007/s10711-014-0013-0
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DOI: https://doi.org/10.1007/s10711-014-0013-0