Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A warped product version of the Cheeger-Gromoll splitting theorem


Author: William Wylie
Journal: Trans. Amer. Math. Soc. 369 (2017), 6661-6681
MSC (2010): Primary 53C20
DOI: https://doi.org/10.1090/tran/7003
Published electronically: May 16, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a new generalization of the Cheeger-Gromoll splitting theorem where we obtain a warped product splitting under the existence of a line. The curvature condition in our splitting is a curvature dimension inequality of the form $ CD(0,1)$. Even though we have to allow warping in our splitting, we are able to recover topological applications. In particular, for a smooth compact Riemannian manifold admitting a density which is $ CD(0,1)$, we show that the fundamental group of $ M$ is the fundamental group of a compact manifold with non-negative sectional curvature. If the space is also locally homogeneous, we obtain that the space also admits a metric of non-negative sectional curvature. Both of these obstructions give many examples of Riemannian metrics which do not admit any smooth density which is $ CD(0,1)$.


References [Enhancements On Off] (What's this?)

  • [Bak94] Dominique Bakry, L'hypercontractivité et son utilisation en théorie des semigroupes, Lectures on probability theory (Saint-Flour, 1992) Lecture Notes in Math., vol. 1581, Springer, Berlin, 1994, pp. 1-114 (French). MR 1307413, https://doi.org/10.1007/BFb0073872
  • [BÉ85] D. Bakry and Michel Émery, Diffusions hypercontractives, Séminaire de probabilités, XIX, 1983/84, Lecture Notes in Math., vol. 1123, Springer, Berlin, 1985, pp. 177-206 (French). MR 889476, https://doi.org/10.1007/BFb0075847
  • [Bor74] Christer Borell, Convex measures on locally convex spaces, Ark. Mat. 12 (1974), 239-252. MR 0388475, https://doi.org/10.1007/BF02384761
  • [BL76] Herm Jan Brascamp and Elliott H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation, J. Functional Analysis 22 (1976), no. 4, 366-389. MR 0450480
  • [CG71] Jeff Cheeger and Detlef Gromoll, The splitting theorem for manifolds of nonnegative Ricci curvature, J. Differential Geometry 6 (1971/72), 119-128. MR 0303460
  • [FLZ09] Fuquan Fang, Xiang-Dong Li, and Zhenlei Zhang, Two generalizations of Cheeger-Gromoll splitting theorem via Bakry-Emery Ricci curvature, Ann. Inst. Fourier (Grenoble) 59 (2009), no. 2, 563-573 (English, with English and French summaries). MR 2521428
  • [FLGRKÜ01] M. Fernández-López, E. García-Río, D. N. Kupeli, and B. Ünal, A curvature condition for a twisted product to be a warped product, Manuscripta Math. 106 (2001), no. 2, 213-217. MR 1865565, https://doi.org/10.1007/s002290100204
  • [Gig14] Nicola Gigli, An overview of the proof of the splitting theorem in spaces with non-negative Ricci curvature, Anal. Geom. Metr. Spaces 2 (2014), 169-213. MR 3210895, https://doi.org/10.2478/agms-2014-0006
  • [Gig] Nicola Gigli, The splitting theorem in non-smooth context. arXiv:1302.5555.
  • [KW17] Lee Kennard and William Wylie, Positive weighted sectional curvature, Indiana Math J. 66 (2017), no. 2, 419-462.
  • [Kla17] Bo'az Klartag, Needle decompositions in Riemannian geometry, Mem. Amer. Math. Soc. 249 (2017), no. 1180, 77 pp.
  • [KM17] Alexander V. Kolesnikov and Emanuel Milman, Brascamp-Lieb-type inequalities on weighted Riemannian manifolds with boundary, J. Geom. Anal. 27 (2017), no. 2, 1680-1702. MR 3625169, https://doi.org/10.1007/s12220-016-9736-5
  • [Lic70] André Lichnerowicz, Variétés riemanniennes à tenseur C non négatif, C. R. Acad. Sci. Paris Sér. A-B 271 (1970), A650-A653 (French). MR 0268812
  • [Lic71] André Lichnerowicz, Variétés kählériennes à première classe de Chern non negative et variétés riemanniennes à courbure de Ricci généralisée non negative, J. Differential Geometry 6 (1971/72), 47-94 (French). MR 0300228
  • [Mil17] Emanuel Milman, Beyond traditional curvature-dimension I: new model spaces for isoperimetric and concentration inequalities in negative dimension, Trans. Amer. Math. Soc. 369 (2017), no. 5, 3605-3637. MR 3605981, https://doi.org/10.1090/tran/6796
  • [Mil] Emanuel Milman, Harmonic measures on the sphere via curvature-dimension, Ann. Fac. Sci. Toulouse Math. arXiv:1505.04335.
  • [Oht16] Shin-ichi Ohta, $ (K,N)$-convexity and the curvature-dimension condition for negative $ N$, J. Geom. Anal. 26 (2016), no. 3, 2067-2096. MR 3511469, https://doi.org/10.1007/s12220-015-9619-1
  • [OT11] Shin-ichi Ohta and Asuka Takatsu, Displacement convexity of generalized relative entropies, Adv. Math. 228 (2011), no. 3, 1742-1787. MR 2824568, https://doi.org/10.1016/j.aim.2011.06.029
  • [OT13] Shin-Ichi Ohta and Asuka Takatsu, Displacement convexity of generalized relative entropies. II, Comm. Anal. Geom. 21 (2013), no. 4, 687-785. MR 3078941, https://doi.org/10.4310/CAG.2013.v21.n4.a1
  • [O'N83] Barrett O'Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
  • [Wil00] Burkhard Wilking, On fundamental groups of manifolds of nonnegative curvature, Differential Geom. Appl. 13 (2000), no. 2, 129-165. MR 1783960, https://doi.org/10.1016/S0926-2245(00)00030-9
  • [WW09] Guofang Wei and Will Wylie, Comparison geometry for the Bakry-Emery Ricci tensor, J. Differential Geom. 83 (2009), no. 2, 377-405. MR 2577473
  • [WW16] Eric Woolgar and William Wylie, Cosmological singularity theorems and splitting theorems for $ N$-Bakry-Émery spacetimes, J. Math. Phys. 57 (2016), no. 2. MR 3453844, https://doi.org/10.1063/1.4940340
  • [Wyl15] William Wylie, Sectional curvature for Riemannian manifolds with density, Geom. Dedicata 178 (2015), 151-169. MR 3397488, https://doi.org/10.1007/s10711-015-0050-3
  • [WY] William Wylie and Dmytro Yeroshkin, On the geometry of Riemannian manifolds with density. arXiv:1602.08000.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C20

Retrieve articles in all journals with MSC (2010): 53C20


Additional Information

William Wylie
Affiliation: Department of Mathematics, 215 Carnegie Building, Syracuse University, Syracuse, New York 13244
Email: wwylie@syr.edu

DOI: https://doi.org/10.1090/tran/7003
Received by editor(s): June 23, 2015
Received by editor(s) in revised form: January 7, 2016, April 4, 2016, and June 16, 2016
Published electronically: May 16, 2017
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society