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Riemannian homogeneous manifold; Einstein manifold; weakly Einstein manifold
Y. Euh, J. Park and K. Sekigawa were the first authors who defined the concept of a weakly Einstein Riemannian manifold as a modification of that of an Einstein Riemannian manifold. The defining formula is expressed in terms of the Riemannian scalar invariants of degree two. This concept was inspired by that of a super-Einstein manifold introduced earlier by A. Gray and T. J. Willmore in the context of mean-value theorems in Riemannian geometry. The dimension $4$ is the most interesting case, where each Einstein space is weakly Einstein. The original authors gave two examples of homogeneous weakly Einstein manifolds (depending on one, or two parameters, respectively) which are not Einstein. The goal of this paper is to prove that these examples are the only existing examples. We use, for this purpose, the classification of $4$-dimensional homogeneous Riemannian manifolds given by L. Bérard Bergery and, also, the basic method and many explicit formulas from our previous article with different topic published in Czechoslovak Math. J. in 2008. We also use Mathematica 7.0 to organize better the tedious routine calculations. The problem of existence of non-homogeneous weakly Einstein spaces in dimension $4$ which are not Einstein remains still unsolved.
[1] Arias-Marco, T., Kowalski, O.: Classification of $4$-dimensional homogeneous D'Atri spaces. Czech. Math. J. 58 (2008), 203-239. DOI 10.1007/s10587-008-0014-y | MR 2402535 | Zbl 1174.53024
[2] Bergery, L. Bérard: Four-dimensional homogeneous Riemannian spaces. Riemannian Geometry in Dimension 4. Papers from the Arthur Besse seminar held at the Université de Paris VII, Paris, 1978/1979 L. Bérard Bergery et al. Mathematical Texts 3 CEDIC, Paris (1981), French. MR 0769130
[3] Boeckx, E., Vanhecke, L.: Unit tangent sphere bundles with constant scalar curvature. Czech. Math. J. 51 (2001), 523-544. DOI 10.1023/A:1013779805244 | MR 1851545 | Zbl 1079.53063
[4] Graaf, W. A. de: Classification of solvable Lie algebras. Exp. Math. 14 (2005), 15-25. DOI 10.1080/10586458.2005.10128911 | MR 2146516 | Zbl 1173.17300
[5] Euh, Y., Park, J., Sekigawa, K.: A curvature identity on a $4$-dimensional Riemannian manifold. Result. Math. 63 (2013), 107-114. DOI 10.1007/s00025-011-0164-3 | MR 3009674 | Zbl 1273.53009
[6] Euh, Y., Park, J., Sekigawa, K.: A generalization of a $4$-dimensional Einstein manifold. Math. Slovaca 63 (2013), 595-610. MR 3071978
[7] Euh, Y., Park, J., Sekigawa, K.: Critical metrics for quadratic functionals in the curvature on $4$-dimensional manifolds. Differ. Geom. Appl. 29 (2011), 642-646. DOI 10.1016/j.difgeo.2011.07.001 | MR 2831820 | Zbl 1228.58010
[8] Gray, A., Willmore, T. J.: Mean-value theorems for Riemannian manifolds. Proc. R. Soc. Edinb., Sect. A 92 (1982), 343-364. DOI 10.1017/S0308210500032571 | MR 0677493 | Zbl 0495.53040
[9] Jensen, G. R.: Homogeneous Einstein spaces of dimension four. J. Differ. Geom. 3 (1969), 309-349. MR 0261487 | Zbl 0194.53203
[10] Milnor, J. W.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21 (1976), 293-329. DOI 10.1016/S0001-8708(76)80002-3 | MR 0425012 | Zbl 0341.53030
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