Three-manifolds with many flat planes

Bettiol, Renato; Schmidt, Benjamin

doi:10.1090/tran/6961

Three-manifolds with many flat planes
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by Renato G. Bettiol and Benjamin Schmidt PDF

Trans. Amer. Math. Soc. 370 (2018), 669-693 Request permission

Abstract:

We discuss the rigidity (or lack thereof) imposed by different notions of having an abundance of zero curvature planes on a complete Riemannian $3$-manifold. We prove a rank rigidity theorem for complete $3$-manifolds, showing that having higher rank is equivalent to having reducible universal covering. We also study $3$-manifolds such that every tangent vector is contained in a flat plane, including examples with irreducible universal covering, and discuss the effect of finite volume and real-analyticity assumptions.

References

Werner Ballmann, Nonpositively curved manifolds of higher rank, Ann. of Math. (2) 122 (1985), no. 3, 597–609. MR 819559, DOI 10.2307/1971331
Keith Burns and Ralf Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 35–59. MR 908215, DOI 10.1007/BF02698934
Jeff Cheeger and Mikhael Gromov, Collapsing Riemannian manifolds while keeping their curvature bounded. I, J. Differential Geom. 23 (1986), no. 3, 309–346. MR 852159
Quo-Shin Chi, A curvature characterization of certain locally rank-one symmetric spaces, J. Differential Geom. 28 (1988), no. 2, 187–202. MR 961513
Christopher Connell, A characterization of homogeneous spaces with positive hyperbolic rank, Geom. Dedicata 93 (2002), 205–233. MR 1934699, DOI 10.1023/A:1020307604978
David Constantine, 2-frame flow dynamics and hyperbolic rank-rigidity in nonpositive curvature, J. Mod. Dyn. 2 (2008), no. 4, 719–740. MR 2449143, DOI 10.3934/jmd.2008.2.719
Patrick Eberlein and Jens Heber, A differential geometric characterization of symmetric spaces of higher rank, Inst. Hautes Études Sci. Publ. Math. 71 (1990), 33–44. MR 1079642, DOI 10.1007/BF02699876
J.-H. Eschenburg and C. Olmos, Rank and symmetry of Riemannian manifolds, Comment. Math. Helv. 69 (1994), no. 3, 483–499. MR 1289339, DOI 10.1007/BF02564499
L. Florit and W. Ziller, Manifolds with conullity at most two as graph manifolds, to appear; arXiv:1611.06572.
M. Gromov, Manifolds of negative curvature, J. Differential Geometry 13 (1978), no. 2, 223–230. MR 540941, DOI 10.4310/jdg/1214434487
Ursula Hamenstädt, A geometric characterization of negatively curved locally symmetric spaces, J. Differential Geom. 34 (1991), no. 1, 193–221. MR 1114460
Michael Kapovich, Flats in 3-manifolds, Ann. Fac. Sci. Toulouse Math. (6) 14 (2005), no. 3, 459–499 (English, with English and French summaries). MR 2172587, DOI 10.5802/afst.1099
Katsumi Nomizu, On local and global existence of Killing vector fields, Ann. of Math. (2) 72 (1960), 105–120. MR 119172, DOI 10.2307/1970148
Benjamin Schmidt, Krishnan Shankar, and Ralf Spatzier, Positively curved manifolds with large spherical rank, Comment. Math. Helv. 91 (2016), no. 2, 219–251. MR 3493370, DOI 10.4171/CMH/384
Benjamin Schmidt and Jon Wolfson, Three-manifolds with constant vector curvature, Indiana Univ. Math. J. 63 (2014), no. 6, 1757–1783. MR 3298721, DOI 10.1512/iumj.2014.63.5436
Kouei Sekigawa, On the Riemannian manifolds of the form $B\times _{f}F$, K\B{o}dai Math. Sem. Rep. 26 (1974/75), 343–347. MR 438253
K. Shankar, R. Spatzier, and B. Wilking, Spherical rank rigidity and Blaschke manifolds, Duke Math. J. 128 (2005), no. 1, 65–81. MR 2137949, DOI 10.1215/S0012-7094-04-12813-1
R. J. Spatzier and M. Strake, Some examples of higher rank manifolds of nonnegative curvature, Comment. Math. Helv. 65 (1990), no. 2, 299–317. MR 1057246, DOI 10.1007/BF02566609
Héctor J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188. MR 321133, DOI 10.1090/S0002-9947-1973-0321133-2
Friedhelm Waldhausen, Eine Klasse von $3$-dimensionalen Mannigfaltigkeiten. I, II, Invent. Math. 3 (1967), 308–333; ibid. 4 (1967), 87–117 (German). MR 235576, DOI 10.1007/BF01402956
Jordan Watkins, The higher rank rigidity theorem for manifolds with no focal points, Geom. Dedicata 164 (2013), 319–349. MR 3054631, DOI 10.1007/s10711-012-9776-3