The structure of submetries
Geometry & topology, Tome 26 (2022) no. 6, pp. 2649-2711.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

We investigate the geometric and topological structure of equidistant decompositions of Riemannian manifolds.

Keywords: Alexandrov space, submetry, singular Riemannian foliation, equidistant decomposition

Kapovitch, Vitali 1 ; Lytchak, Alexander 2

1 Department of Mathematics, University of Toronto, Toronto, ON, Canada
2 Mathematisches Institut, Universität Köln, Köln, Germany
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Kapovitch, Vitali; Lytchak, Alexander. The structure of submetries. Geometry & topology, Tome 26 (2022) no. 6, pp. 2649-2711. http://geodesic.mathdoc.fr/item/GT_2022_26_6_a4/

[1] S Alexander, R L Bishop, FK–convex functions on metric spaces, Manuscripta Math. 110 (2003) 115 | DOI

[2] S Alexander, V Kapovitch, A Petrunin, Alexandrov geometry, preprint (2019)

[3] M M Alexandrino, M Radeschi, Smoothness of isometric flows on orbit spaces and applications, Transform. Groups 22 (2017) 1 | DOI

[4] L Ambrosio, Calculus, heat flow and curvature-dimension bounds in metric measure spaces, from: "Proceedings of the International Congress of Mathematicians, I" (editors B Sirakov, P N d Souza, M Viana), World Sci. (2018) 301

[5] L Ambrosio, J Bertrand, DC calculus, Math. Z. 288 (2018) 1037 | DOI

[6] L Ambrosio, N Gigli, G Savaré, Metric measure spaces with Riemannian Ricci curvature bounded from below, Duke Math. J. 163 (2014) 1405 | DOI

[7] L Ambrosio, B Kirchheim, Rectifiable sets in metric and Banach spaces, Math. Ann. 318 (2000) 527 | DOI

[8] V Bangert, Sets with positive reach, Arch. Math. (Basel) 38 (1982) 54 | DOI

[9] V N Berestovskii, Submetries of three-dimensional forms of nonnegative curvature, Sibirsk. Mat. Zh. 28 (1987) 44

[10] V N Berestovskii, Generalized normal homogeneous spheres, Sibirsk. Mat. Zh. 54 (2013) 742

[11] V N Berestovskii, L Guijarro, A metric characterization of Riemannian submersions, Ann. Global Anal. Geom. 18 (2000) 577 | DOI

[12] V N Berestovskii, I G Nikolaev, Multidimensional generalized Riemannian spaces, from: "Geometry, IV" (editor Y G Reshetnyak), Encycl. Math. Sci. 70, Springer (1993) 165 | DOI

[13] V N Berestovskii, Y G Nikonorov, Finite homogeneous metric spaces, Sibirsk. Mat. Zh. 60 (2019) 973 | DOI

[14] G E Bredon, Introduction to compact transformation groups, 46, Academic (1972)

[15] W Browder, Higher torsion in H–spaces, Trans. Amer. Math. Soc. 108 (1963) 353 | DOI

[16] Y Burago, M Gromov, G Perelman, A D Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk 47 (1992) 3

[17] J Cheeger, K Fukaya, M Gromov, Nilpotent structures and invariant metrics on collapsed manifolds, J. Amer. Math. Soc. 5 (1992) 327 | DOI

[18] X Chen, K Grove, Rigidity theorems for submetries in positive curvature, Adv. Math. 289 (2016) 784 | DOI

[19] E Dyer, M E Hamstrom, Completely regular mappings, Fund. Math. 45 (1958) 103 | DOI

[20] H Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959) 418 | DOI

[21] S C Ferry, Alexander duality and Hurewicz fibrations, Trans. Amer. Math. Soc. 327 (1991) 201 | DOI

[22] T Foertsch, A Lytchak, The de Rham decomposition theorem for metric spaces, Geom. Funct. Anal. 18 (2008) 120 | DOI

[23] T Fujioka, Regular points of extremal subsets in Alexandrov spaces, preprint (2019)

[24] F Galaz-García, L Guijarro, Isometry groups of Alexandrov spaces, Bull. Lond. Math. Soc. 45 (2013) 567 | DOI

[25] F Galaz-García, M Kell, A Mondino, G Sosa, On quotients of spaces with Ricci curvature bounded below, J. Funct. Anal. 275 (2018) 1368 | DOI

[26] F Galaz-García, M Radeschi, Singular Riemannian foliations and applications to positive and non-negative curvature, J. Topol. 8 (2015) 603 | DOI

[27] J Ge, N Li, Gluing of multiple Alexandrov spaces, preprint (2020)

[28] J Ge, M Radeschi, Differentiable classification of 4–manifolds with singular Riemannian foliations, Math. Ann. 363 (2015) 525 | DOI

[29] D Gromoll, K Grove, A generalization of Berger’s rigidity theorem for positively curved manifolds, Ann. Sci. École Norm. Sup. 20 (1987) 227 | DOI

[30] K Grove, A Moreno, P Petersen, The boundary conjecture for leaf spaces, Ann. Inst. Fourier (Grenoble) 69 (2019) 2941

[31] K Grove, F Wilhelm, Metric constraints on exotic spheres via Alexandrov geometry, J. Reine Angew. Math. 487 (1997) 201 | DOI

[32] L Guijarro, V Kapovitch, Restrictions on the geometry at infinity of nonnegatively curved manifolds, Duke Math. J. 78 (1995) 257 | DOI

[33] L Guijarro, G Walschap, Submetries vs. submersions, Rev. Mat. Iberoam. 27 (2011) 605 | DOI

[34] L Guijarro, F Wilhelm, Focal radius, rigidity, and lower curvature bounds, Proc. Lond. Math. Soc. 116 (2018) 1519 | DOI

[35] J Harvey, C Searle, Orientation and symmetries of Alexandrov spaces with applications in positive curvature, J. Geom. Anal. 27 (2017) 1636 | DOI

[36] V Kapovitch, Regularity of limits of noncollapsing sequences of manifolds, Geom. Funct. Anal. 12 (2002) 121 | DOI

[37] V Kapovitch, Restrictions on collapsing with a lower sectional curvature bound, Math. Z. 249 (2005) 519 | DOI

[38] V Kapovitch, A Lytchak, Remarks on manifolds with two-sided curvature bounds, Anal. Geom. Metr. Spaces 9 (2021) 53 | DOI

[39] V Kapovitch, A Lytchak, A Petrunin, Metric-measure boundary and geodesic flow on Alexandrov spaces, J. Eur. Math. Soc. 23 (2021) 29 | DOI

[40] M Karmanova, Rectifiable sets and coarea formula for metric-valued mappings, J. Funct. Anal. 254 (2008) 1410 | DOI

[41] N Kleinjohann, Nächste Punkte in der Riemannschen Geometrie, Math. Z. 176 (1981) 327 | DOI

[42] C Lange, Orbifolds from a metric viewpoint, Geom. Dedicata 209 (2020) 43 | DOI

[43] N Lebedeva, A Nepechiy, Alexandrov regions, in preparation

[44] N Lebedeva, A Petrunin, V Zolotov, Bipolar comparison, Geom. Funct. Anal. 29 (2019) 258 | DOI

[45] N Li, A Naber, Quantitative estimates on the singular sets of Alexandrov spaces, Peking Math. J. 3 (2020) 203 | DOI

[46] A Lytchak, Allgemeine Theorie der Submetrien und verwandte mathematische Probleme, , Universität Bonn, Mathematisches Institut (2002)

[47] A Lytchak, Differentiation in metric spaces, Algebra i Analiz 16 (2004) 128

[48] A Lytchak, On the geometry of subsets of positive reach, Manuscripta Math. 115 (2004) 199 | DOI

[49] A Lytchak, Almost convex subsets, Geom. Dedicata 115 (2005) 201 | DOI

[50] A Lytchak, Open map theorem for metric spaces, Algebra i Analiz 17 (2005) 139

[51] A Lytchak, Rigidity of spherical buildings and joins, Geom. Funct. Anal. 15 (2005) 720 | DOI

[52] A Lytchak, K Nagano, Topological regularity of spaces with an upper curvature bound, J. Eur. Math. Soc. 24 (2022) 137 | DOI

[53] A Lytchak, G Thorbergsson, Curvature explosion in quotients and applications, J. Differential Geom. 85 (2010) 117

[54] A Lytchak, A Yaman, On Hölder continuous Riemannian and Finsler metrics, Trans. Amer. Math. Soc. 358 (2006) 2917 | DOI

[55] J Mather, Notes on topological stability, Bull. Amer. Math. Soc. 49 (2012) 475 | DOI

[56] R A E Mendes, M Radeschi, A slice theorem for singular Riemannian foliations, with applications, Trans. Amer. Math. Soc. 371 (2019) 4931 | DOI

[57] R A E Mendes, M Radeschi, Laplacian algebras, manifold submetries and the inverse invariant theory problem, Geom. Funct. Anal. 30 (2020) 536 | DOI

[58] P Molino, Riemannian foliations, 73, Birkhäuser (1988) | DOI

[59] A Nepechiy, Toward canonical convex functions in Alexandrov spaces, preprint (2019)

[60] G Perelman, Alexandrov’s spaces with curvature bounded below, II, preprint (1991)

[61] G Perelman, DC structure on Alexandrov space, preprint (1994)

[62] G Perelman, Proof of the soul conjecture of Cheeger and Gromoll, J. Differential Geom. 40 (1994) 209

[63] G Y Perelman, A M Petrunin, Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, Algebra i Analiz 5 (1993) 242

[64] G Perelman, A Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, preprint (1995)

[65] A Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8 (1998) 123 | DOI

[66] A Petrunin, Semiconcave functions in Alexandrov’s geometry, from: "Surveys in differential geometry, XI : Metric and comparison geometry" (editors J Cheeger, K Grove), Surv. Differ. Geom. 11, International (2007) 137 | DOI

[67] M Radeschi, Clifford algebras and new singular Riemannian foliations in spheres, Geom. Funct. Anal. 24 (2014) 1660 | DOI

[68] J Rataj, L Zajíček, On the structure of sets with positive reach, Math. Nachr. 290 (2017) 1806 | DOI

[69] F Raymond, Local triviality for Hurewicz fiberings of manifolds, Topology 3 (1965) 43 | DOI

[70] T Shioya, T Yamaguchi, Collapsing three-manifolds under a lower curvature bound, J. Differential Geom. 56 (2000) 1

[71] G Thorbergsson, Singular Riemannian foliations and isoparametric submanifolds, Milan J. Math. 78 (2010) 355 | DOI

[72] G S Ungar, Conditions for a mapping to have the slicing structure property, Pacific J. Math. 30 (1969) 549 | DOI

[73] B Wilking, A duality theorem for Riemannian foliations in nonnegative sectional curvature, Geom. Funct. Anal. 17 (2007) 1297 | DOI

[74] T Yamaguchi, Collapsing and pinching under a lower curvature bound, Ann. of Math. 133 (1991) 317 | DOI

[75] T Yamaguchi, Collapsing 4–manifolds under a lower curvature bound, preprint (2012)