Positively curved Killing foliations via deformations

Caramello, Francisco, Jr.; Töben, Dirk

doi:10.1090/tran/7893

Positively curved Killing foliations via deformations
HTML articles powered by AMS MathViewer

by Francisco C. Caramello Jr. and Dirk Töben PDF

Trans. Amer. Math. Soc. 372 (2019), 8131-8158 Request permission

Abstract:

We show that a compact manifold that admits a Killing foliation with positive transverse curvature fibers over finite quotients of spheres or weighted complex projective spaces provided that the singular foliation defined by the closures of the leaves has maximal dimension. This result is obtained by deforming the foliation into a closed one while maintaining transverse geometric properties, which allows us to apply results from the Riemannian geometry of orbifolds to the space of leaves. We also show that the basic Euler characteristic is preserved by such deformations. Using this fact, we prove that a Riemannian foliation of a compact manifold with finite fundamental group and nonvanishing Euler characteristic is closed. As another application, we obtain that, for a positively curved Killing foliation of a compact manifold, if the structural algebra has sufficiently large dimension, then the basic Euler characteristic is positive.

References

Alejandro Adem, Johann Leida, and Yongbin Ruan, Orbifolds and stringy topology, Cambridge Tracts in Mathematics, vol. 171, Cambridge University Press, Cambridge, 2007. MR 2359514, DOI 10.1017/CBO9780511543081
Aziz El Kacimi-Alaoui, Vlad Sergiescu, and Gilbert Hector, La cohomologie basique d’un feuilletage riemannien est de dimension finie, Math. Z. 188 (1985), no. 4, 593–599 (French). MR 774559, DOI 10.1007/BF01161658
Marcos M. Alexandrino and Renato G. Bettiol, Lie groups and geometric aspects of isometric actions, Springer, Cham, 2015. MR 3362465, DOI 10.1007/978-3-319-16613-1
C. Allday and V. Puppe, Cohomological methods in transformation groups, Cambridge Studies in Advanced Mathematics, vol. 32, Cambridge University Press, Cambridge, 1993. MR 1236839, DOI 10.1017/CBO9780511526275
A. V. Bagaev and N. I. Zhukova, The isometry groups of Riemannian orbifolds, Sibirsk. Mat. Zh. 48 (2007), no. 4, 723–741 (Russian, with Russian summary); English transl., Siberian Math. J. 48 (2007), no. 4, 579–592. MR 2355369, DOI 10.1007/s11202-007-0060-y
Victor Belfi, Efton Park, and Ken Richardson, A Hopf index theorem for foliations, Differential Geom. Appl. 18 (2003), no. 3, 319–341. MR 1975032, DOI 10.1016/S0926-2245(02)00165-1
Joseph E. Borzellino, Orbifolds of maximal diameter, Indiana Univ. Math. J. 42 (1993), no. 1, 37–53. MR 1218706, DOI 10.1512/iumj.1993.42.42004
Joseph E. Borzellino and Victor Brunsden, The stratified structure of spaces of smooth orbifold mappings, Commun. Contemp. Math. 15 (2013), no. 5, 1350018, 37. MR 3117354, DOI 10.1142/S0219199713500181
Charles P. Boyer and Krzysztof Galicki, Sasakian geometry, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2008. MR 2382957
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Alberto Candel and Lawrence Conlon, Foliations. I, Graduate Studies in Mathematics, vol. 23, American Mathematical Society, Providence, RI, 2000. MR 1732868, DOI 10.1090/gsm/023
Weimin Chen and Yongbin Ruan, Orbifold Gromov-Witten theory, Orbifolds in mathematics and physics (Madison, WI, 2001) Contemp. Math., vol. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 25–85. MR 1950941, DOI 10.1090/conm/310/05398
P. E. Conner, On the action of the circle group, Michigan Math. J. 4 (1957), 241–247. MR 96747, DOI 10.1307/mmj/1028997955
Theodore Frankel, Manifolds with positive curvature, Pacific J. Math. 11 (1961), 165–174. MR 123272, DOI 10.2140/pjm.1961.11.165
Fernando Galaz-García, Martin Kerin, Marco Radeschi, and Michael Wiemeler, Torus orbifolds, slice-maximal torus actions, and rational ellipticity, Int. Math. Res. Not. IMRN 18 (2018), 5786–5822. MR 3862119, DOI 10.1093/imrn/rnx064
Étienne Ghys, Feuilletages riemanniens sur les variétés simplement connexes, Ann. Inst. Fourier (Grenoble) 34 (1984), no. 4, 203–223 (French, with English summary). MR 766280, DOI 10.5802/aif.994
Oliver Goertsches and Dirk Töben, Equivariant basic cohomology of Riemannian foliations, J. Reine Angew. Math. 745 (2018), 1–40. MR 3881470, DOI 10.1515/crelle-2015-0102
Alexander Gorokhovsky and John Lott, The index of a transverse Dirac-type operator: the case of abelian Molino sheaf, J. Reine Angew. Math. 678 (2013), 125–162. MR 3056105, DOI 10.1515/crelle.2012.020
Detlef Gromoll and Karsten Grove, The low-dimensional metric foliations of Euclidean spheres, J. Differential Geom. 28 (1988), no. 1, 143–156. MR 950559
Karsten Grove and Catherine Searle, Positively curved manifolds with maximal symmetry-rank, J. Pure Appl. Algebra 91 (1994), no. 1-3, 137–142. MR 1255926, DOI 10.1016/0022-4049(94)90138-4
A. Haefliger, Groupoïdes d’holonomie et classifiants, Astérisque 116 (1986), 321–331.
A. Haefliger and É. Salem, Riemannian foliations on simply connected manifolds and actions of tori on orbifolds, Illinois J. Math. 34 (1990), no. 4, 706–730. MR 1062771, DOI 10.1215/ijm/1255988064
John Harvey and Catherine Searle, Orientation and symmetries of Alexandrov spaces with applications in positive curvature, J. Geom. Anal. 27 (2017), no. 2, 1636–1666. MR 3625167, DOI 10.1007/s12220-016-9734-7
James J. Hebda, Curvature and focal points in Riemannian foliations, Indiana Univ. Math. J. 35 (1986), no. 2, 321–331. MR 833397, DOI 10.1512/iumj.1986.35.35019
Lee Kennard, On the Hopf conjecture with symmetry, Geom. Topol. 17 (2013), no. 1, 563–593. MR 3039770, DOI 10.2140/gt.2013.17.563
Bruce Kleiner and John Lott, Geometrization of three-dimensional orbifolds via Ricci flow, Astérisque 365 (2014), 101–177 (English, with English and French summaries). MR 3244330
Shoshichi Kobayashi, Transformation groups in differential geometry, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1972 edition. MR 1336823
Jesús A. Alvarez López, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom. 10 (1992), no. 2, 179–194. MR 1175918, DOI 10.1007/BF00130919
Ernesto Lupercio and Bernardo Uribe, Gerbes over orbifolds and twisted $K$-theory, Comm. Math. Phys. 245 (2004), no. 3, 449–489. MR 2045679, DOI 10.1007/s00220-003-1035-x
I. Moerdijk and J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge Studies in Advanced Mathematics, vol. 91, Cambridge University Press, Cambridge, 2003. MR 2012261, DOI 10.1017/CBO9780511615450
Xiaochun Rong and Xiaole Su, The Hopf conjecture for manifolds with abelian group actions, Commun. Contemp. Math. 7 (2005), no. 1, 121–136. MR 2129791, DOI 10.1142/S0219199705001660
Pierre Molino, Désingularisation des feuilletages Riemanniens, Amer. J. Math. 106 (1984), no. 5, 1091–1106 (French). MR 761580, DOI 10.2307/2374274
Pierre Molino, Riemannian foliations, Progress in Mathematics, vol. 73, Birkhäuser Boston, Inc., Boston, MA, 1988. Translated from the French by Grant Cairns; With appendices by Cairns, Y. Carrière, É. Ghys, E. Salem and V. Sergiescu. MR 932463, DOI 10.1007/978-1-4684-8670-4
Witold Mozgawa, Feuilletages de Killing, Collect. Math. 36 (1985), no. 3, 285–290 (French, with English summary). MR 868544
Gen-ichi Oshikiri, On transverse Killing fields of metric foliations of manifolds with positive curvature, Manuscripta Math. 104 (2001), no. 4, 527–531. MR 1836112, DOI 10.1007/s002290170025
Peter Petersen, Riemannian geometry, 2nd ed., Graduate Texts in Mathematics, vol. 171, Springer, New York, 2006. MR 2243772
Thomas Püttmann and Catherine Searle, The Hopf conjecture for manifolds with low cohomogeneity or high symmetry rank, Proc. Amer. Math. Soc. 130 (2002), no. 1, 163–166. MR 1855634, DOI 10.1090/S0002-9939-01-06039-7
E. Salem, Riemannian foliations and pseudogroups of isometries, Appendix D in P. Molino, Riemannian foliations, Progress in Mathematics, vol. 73, Birkhäuser Boston, Inc., Boston, MA, 1988, pp. 265–296.
I. Satake, On a generalization of the notion of manifold, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363. MR 79769, DOI 10.1073/pnas.42.6.359
Dmytro Yeroshkin, Riemannian orbifolds with non-negative curvature, ProQuest LLC, Ann Arbor, MI, 2014. Thesis (Ph.D.)–University of Pennsylvania. MR 3251232