Abstract
We study global obstructions to the eigenvalues of the Ricci tensor on a Riemannian 3-manifold. As a topological obstruction, we first show that if the 3-manifold is closed, then certain choices of the eigenvalues are prohibited: in particular, there is no Riemannian metric whose corresponding Ricci eigenvalues take the form (−μ, f, f), where μ is a positive constant and f is a smooth positive function. We then concentrate on the case when one of the eigenvalues is zero. Here we show that if the manifold is complete and its Ricci eigenvalues take the form (0, λ, λ), where λ is a positive constant, then its universal cover must split isometrically. If the manifold is closed, scalar-flat, and its zero eigenspace contains a unit length vector field that is geodesic and divergence-free, then the manifold must be flat. Our techniques also apply to the study of Ricci solitons in dimension three.
Communicated by: P. Eberlein
Funding: This work was supported by the World Premier International Research Center Initiative (WPI), MEXT, Japan; this manuscript first appeared when both authors were members of the Kavli Institute for the Physics and Mathematics of the Universe (IPMU), University of Tokyo, Japan.
Acknowledgements
We kindly thank Benjamin Schmidt, Wolfgang Ziller, and Robert Bryant for very helpful discussions.
References
[1] A. B. Aazami, The Newman–Penrose formalism for Riemannian 3-manifolds. J. Geom. Phys. 94 (2015), 1–7. MR3350264 Zbl 1319.5302310.1016/j.geomphys.2015.03.009Search in Google Scholar
[2] R. G. Bettiol, B. Schmidt, Three-manifolds with many flat planes. Trans. Amer. Math. Soc. 370 (2018), 669–693. MR3717993 Zbl 1377.5302110.1090/tran/6961Search in Google Scholar
[3] C. H. Brans, Some restrictions on algebraically general vacuum metrics. J. Mathematical Phys. 16 (1975), 1008–1010. MR0408700 Zbl 0304.5302210.1063/1.522621Search in Google Scholar
[4] H.-D. Cao, Recent progress on Ricci solitons. In: Recent advances in geometric analysis, volume 11 of Adv. Lect. Math., 1–38, Int. Press, Somerville, MA 2010. MR2648937 Zbl 1201.53046Search in Google Scholar
[5] O. R. Kowalski, F. Prüfer, On Riemannian 3-manifolds with distinct constant Ricci eigenvalues. Math. Ann. 300 (1994), 17–28. MR1289828 Zbl 0813.5302010.1007/BF01450473Search in Google Scholar
[6] J. M. Lee, Introduction to smooth manifolds. Springer 2013. MR2954043 Zbl 1258.53002Search in Google Scholar
[7] J. Milnor, Curvatures of left invariant metrics on Lie groups. Advances in Math. 21 (1976), 293–329. MR0425012 Zbl 0341.5303010.1016/S0001-8708(76)80002-3Search in Google Scholar
[8] E. Newman, R. Penrose, An approach to gravitational radiation by a method of spin coefficients. J. Mathematical Phys. 3 (1962), 566–578. MR0141500 Zbl 0108.4090510.1063/1.1724257Search in Google Scholar
[9] B. O’Neill, The geometry of Kerr black holes. A K Peters, Ltd., Wellesley, MA 1995. MR1328643 Zbl 0828.53078Search in Google Scholar
[10] P. Petersen, Riemannian geometry. Springer 2006. MR2243772 Zbl 1220.53002Search in Google Scholar
[11] B. Schmidt, J. Wolfson, Three-manifolds with constant vector curvature. Indiana Univ. Math. J. 63 (2014), 1757–1783. MR3298721 Zbl 1315.5304010.1512/iumj.2014.63.5436Search in Google Scholar
[12] A. Spiro, F. Tricerri, 3-dimensional Riemannian metrics with prescribed Ricci principal curvatures. J. Math. Pures Appl. (9) 74 (1995), 253–271. MR1327884 Zbl 0851.53022Search in Google Scholar
[13] Z. I. Szabó, Structure theorems on Riemannian spaces satisfying R(X, Y) ⋅ R = 0 II. Global versions. Geom. Dedicata19 (1985), 65–108. MR797152 Zbl 0612.5302310.1007/BF00233102Search in Google Scholar
[14] C. H. Taubes, The Seiberg–Witten equations and the Weinstein conjecture. Geom. Topol. 11 (2007), 2117–2202. MR2350473 Zbl 1135.5701510.2140/gt.2007.11.2117Search in Google Scholar
[15] K. Yamato, A characterization of locally homogeneous Riemann manifolds of dimension 3. Nagoya Math. J. 123 (1991), 77–90. MR1126183 Zbl 0738.5303210.1017/S0027763000003652Search in Google Scholar
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