Abstract
In this paper we study sectional curvature bounds for Riemannian manifolds with density from the perspective of a weighted torsion-free connection introduced recently by the last two authors. We develop two new tools for studying weighted sectional curvature bounds: a new weighted Rauch comparison theorem and a modified notion of convexity for distance functions. As applications we prove generalizations of theorems of Preissman and Byers for negative curvature, the (homeomorphic) quarter-pinched sphere theorem, and Cheeger’s finiteness theorem. We also improve results of the first two authors for spaces of positive weighted sectional curvature and symmetry.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Berard-Bergery, L.: Les variétés riemanniennes homogènes simplement connexes de dimension impaire à courbure strictement positive. J. Math. Pures Appl. (9) 55(1), 47–67 (1976)
Burago, D., Ivanov, S.: Riemannian tori without conjugate points are flat. Geom. Funct. Anal. 4(3), 259–269 (1994)
Byers, W.P.: On a theorem of Preissmann. Proc. Am. Math. Soc. 24, 50–51 (1970)
Croke, C., Schroeder, V.: The fundamental group of compact manifolds without conjugate points. Comment. Math. Helv. 61(1), 161–175 (1986)
do Carmo, M.P.: Riemannian geometry. In: Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston (1992). Translated from the second Portuguese edition by Francis Flaherty (1992)
Dessai, A., Wilking, B.: Torus actions on homotopy complex projective spaces. Math. Z. 247, 505–511 (2004)
Eschenburg, J.-H., Kerin, M.: Almost positive curvature on the Gromoll-Meyer sphere. Proc. Am. Math. Soc. 136(9), 3263–3270 (2008)
Fang, F., Grove, K.: Reflection groups in non-negative curvature. J. Differ. Geom. 102(2), 179–205 (2016)
Fang, F., Rong, X.: Homeomorphism classification of positively curved manifolds with almost maximal symmetry rank. Math. Ann. 332, 81–101 (2005)
Grove, K.: Critical point theory for distance functions. In: Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), pp. 357–385 (1993)
Grove, K., Searle, C.: Positively curved manifolds with maximal symmetry rank. J. Pure Appl. Algebra 91(1), 137–142 (1994)
Grove, K., Searle, C.: Differential topological restrictions curvature and symmetry. J. Differ. Geom. 47(3), 530–559 (1997)
Heintze, E., Karcher, H.: A general comparison theorem with applications to volume estimates for submanifolds. Ann. Sci. École Norm. Sup. (4) 11(4), 451–470 (1978)
Ivanov, S., Kapovitch, V.: Manifolds without conjugate points and their fundamental groups. J. Differ. Geom. 96(2), 223–240 (2014)
Kennard, L., Wylie, W.: Positive weighted sectional curvature. Indiana Univ. Math. J. 66(2), 419–462 (2017)
Li, J., Xia, C.: An integral formula for affine connections. J. Geom. Anal. 27(3), 2539–2556 (2017)
Petersen, P.: Riemannian Geometry. Third Graduate Texts in Mathematics, vol. 171. Springer, Cham (2016)
Petersen, P.: Convergence theorems in Riemannian geometry. In: Comparison Geometry (Berkeley, CA, 1993–1994), pp. 167–202 (1997)
Petersen, P., Wilhelm, F.: An exotic sphere with positive curvature. Preprint. arXiv:0805.0812v3
Sakurai, Y.: Comparison geometry of manifolds with boundary under a lower weighted Ricci curvature bound. arXiv:1612.08483
Wallach, N.R.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. 96(2), 277–295 (1972)
Wilhelm, F.: An exotic sphere with positive curvature almost everywhere. J. Geom. Anal. 11(3), 519–560 (2001)
Wilking, B.: Manifolds with positive sectional curvature almost everywhere. Invent. Math. 148(1), 117–141 (2002)
Wilking, B.: Torus actions on manifolds of positive sectional curvature. Acta Math. 191(2), 259–297 (2003)
Woolgar, E., Wylie, W.: Cosmological singularity theorems and splitting theorems for N-Bakry–Émery spacetimes. J. Math. Phys. 57(2), 022504 (2016)
Woolgar, E., Wylie, W.: Curvature-dimension bounds for Lorentzian splitting theorems. Preprint. arXiv:1707.09058
Wylie, W.: Sectional curvature for Riemannian manifolds with density. Geom. Dedicata 178, 151–169 (2015)
Wylie, W.: Some curvature pinching results for Riemannian manifolds with density. Proc. Am. Math. Soc. 144(2), 823–836 (2016)
Wylie, W.: A warped product version of the Cheeger–Gromoll splitting theorem. Trans. Am. Math. Soc. 369(9), 6661–6681 (2017)
Wylie, W., Yeroshkin, D.: On the geometry of Riemannian manifolds with density. Preprint. arXiv:1602.08000
Wilking, B., Ziller, W.: Revisiting homogeneous spaces with positive curvature. J. Reine Angew. Math. (to appear). arXiv:1503.06256
Acknowledgements
This work was partially supported by NSF Grant DMS-1440140 while Lee Kennard and William Wylie were in residence at MSRI in Berkeley, California, during the Spring 2016 semester. Lee Kennard was partially supported by NSF Grant DMS-1622541. William Wylie was supported by a grant from the Simons Foundation (#355608, William Wylie) and a grant from the National Science Foundation (DMS-1654034). Dmytro Yeroshkin was partially supported by a grant from the College of Science and Engineering at Idaho State University. We would like to thank the referee for a thorough reading of the paper and many suggestions that improve the readability of the text.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kennard, L., Wylie, W. & Yeroshkin, D. The Weighted Connection and Sectional Curvature for Manifolds With Density. J Geom Anal 29, 957–1001 (2019). https://doi.org/10.1007/s12220-018-0025-3
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-018-0025-3