Abstract
In this paper, we introduce the notion of weakly \(\eta \)-Einstein structure. Then we prove that a 3-dimensional \(\eta \)-Einstein almost contact metric manifold is weakly \(\eta \)-Einstein. Moreover, the generalized Sasakian space forms are weakly \(\eta \)-Einstein. Furthermore, we obtain the characteristic equation for a non-Sasakian contact \((k,\mu )\)-space to be weakly \(\eta \)-Einstein, which provides many interesting examples. In particular, we determine the base manifold whose unit tangent sphere bundle \(T_1M(c)\) is weakly \(\eta \)-Einstein.
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References
Alegre, P., Blair, D.E., Carriazo, A.: Generalized Sasakian-space-forms Israel. J. Math. 141, 157–183 (2004)
Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds, 2nd edn. Birkhäuser Boston Inc, Boston, MA (2010)
Blair, D.E., Koufogiorgos, T., Papantoniou, B.J.: Contact metric manifolds satisfying a nullity condition Israel. J. Math. 91, 189–214 (1995)
Boeckx, E.: A full classification of contact metric \((k,\mu )\)-spaces Illinois. J. Math. 44, 212–219 (2000)
Boeckx, E., Vanhecke, L.: Characteristic reflections on unit tangent sphere bundles Houston. J. Math. 23, 427–448 (1997)
Chen, X.: On weakly Einstein almost contact manifolds. J. Korean Math. Soc. 57, 707–719 (2020)
Cho, J.T.: Contact hypersurfaces and CR-symmetry. Ann. Mat. Pura Appl. 199, 1873–1884 (2020)
Euh, Y., Park, J.H., Sekigawa, K.: A curvature identity on a 4-dimensional Riemannian manifold. Res. Math. 63, 107–114 (2013)
Gray, A., Willmore, T.J.: Mean-value theorems for Riemannian manifolds. Proc. R. Soc. Edinb. Sect. A 92(3–4), 343–364 (1982)
Park, J.H., Sekigawa, K.: When are the tangent sphere bundles of a Riemannian manifold eta-Einstein? Ann. Glob. Anal. Geom. 36, 275–284 (2009)
SasakiOn, S.: The differential geometry of tangent bundle of Riemannian manifolds. Tohoku Math. J. 10, 338–354 (1958)
Tanno, S.: The topology of contact Riemannian manifolds Illinois. J. Math. 12, 700–717 (1968)
Funding
J. T. Cho was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2019R1F1A1040829) and S. H. Chun was supported by the National Research Foundation of Korea(NRF) Grant funded by the Korea government(MSIT) (No. 2021R1F1A1055679).
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Cho, J.T., Chun, S.H. & Euh, Y. Weakly \(\eta \)-Einstein Contact Manifolds. Results Math 77, 110 (2022). https://doi.org/10.1007/s00025-022-01645-0
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DOI: https://doi.org/10.1007/s00025-022-01645-0
Keywords
- Weakly \(\eta \)-Einstein
- \(\text {(}k</Keyword> <Keyword>\mu \text {)}\)-space
- unit tangent sphere bundle