Abstract
It is shown that locally conformally flat weakly-Einstein manifolds are either locally symmetric, and hence a product \(N_1^m(c)\times N_2^m(-c)\), or otherwise they are locally homothetic to some specific warped product metrics \({\mathcal {I}}\times _fN(c)\). As an application we classify weakly-Einstein hypersurfaces in the Euclidean space.
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References
Arias-Marco, T., Kowalski, O.: Classification of 4-dimensional homogeneous weakly Einstein manifolds. Czech. Math. J. 65, 21–59 (2015)
Besse, A.L.: Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete 10. Springer, Berlin (1987)
Besse, A.L.: Manifolds all of whose Geodesics are Closed, Ergebnisse der Mathematik und ihrer Grenzgebiete 93. Springer, Berlin, New York (1978)
Bivens, I., Bourguignon, J.-P., Derdziński, A., Ferus, D., Kowalski, O., Milnor, T., Oliker, V., Simon, U., Strübing, W., Voss, K.: Discussion on Codazzi-tensors Global differential geometry and global analysis (Berlin, 1979), 243–299, Lecture Notes in Math., 838, Springer, Berlin, New York (1981)
Boeckx, E., Vanhecke, L.: Unit tangent sphere bundles with constant scalar curvature. Czech. Math. J. 51(126), 523–544 (2001)
Brozos-Vázquez, M., García-Río, E., Vázquez-Lorenzo, R.: Complete locally conformally flat manifolds of negative curvature. Pac. J. Math. 226, 201–219 (2006)
Cecil, T.E., Ryan, P.J.: Geometry of Hypersurfaces. Monographs in Mathematics. Springer, New York (2015)
Chen, B.Y., Vanhecke, L.: Differential geometry of geodesic spheres. J. Reine Angew. Math. 325, 28–67 (1981)
Euh, Y., Park, J., Sekigawa, K.: A curvature identity on a 4-dimensional Riemannian manifold. Result. Math. 63, 107–114 (2013)
Euh, Y., Park, J., Sekigawa, K.: Critical metrics for quadratic functionals in the curvature on 4-dimensional manifolds. Differ. Geom. Appl. 29, 642–646 (2011)
Fialkow, A.: Hypersurfaces of a space of constant curvature. Ann. Math. (2) 39, 762–785 (1938)
Gray, A., Willmore, T.J.: Mean-value theorems for Riemannian manifolds. Proc. R. Soc. Edinb. Sect. A 92, 343–364 (1982)
Hwang, S., Yun, G.: Weakly Einstein critical point equation. Bull. Korean Math. Soc. 53, 1087–1094 (2016)
Merton, G.: Codazzi tensors with two eigenvalue functions. Proc. Am. Math. Soc. 141, 3265–3273 (2013)
Nishikawa, S., Maeda, Y.: Conformally flat hypersurfaces in a conformally flat Riemannian manifold. Tohoku Math. J. 26, 159–168 (1974)
O’Neill, B.: Semi-Riemannian Geometry, with Applications to Relativity. Academic, New York (1983)
Takagi, H.: Conformally flat Riemannian manifolds admitting a transitive group of isometries. Tohoku Math. J. (2) 27, 103–110 (1975)
Yau, S.T.: Remarks on conformal transformations. J. Differ. Geom. 8, 369–381 (1973)
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Supported by projects ED431F 2017/03, and MTM2016-75897-P (Spain). The work on this project was done whilst the second author was visiting University of Santiago de Compostela in Spain during Sept 2017-June 2018. This visit was funded by University of Bonab.
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García-Río, E., Haji-Badali, A., Mariño-Villar, R. et al. Locally conformally flat weakly-Einstein manifolds. Arch. Math. 111, 549–559 (2018). https://doi.org/10.1007/s00013-018-1221-x
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DOI: https://doi.org/10.1007/s00013-018-1221-x