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On the Pseudohermitian Curvature of Contact Semi-Riemannian Manifolds

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Abstract

Let M be a contact semi-Riemannian manifold, equivalently a non degenerate almost CR manifold. In this paper we study the pseudo-hermitian Ricci curvature, pseudo-Einstein and \(\eta \)-Einstein manifolds. Then, by using the pseudo-Einstein and the \(\eta \)-Einstein conditions, some rigidity theorems are established to characterize Sasakian manifolds among nondegenerate CR manifolds. In particular, if the Webster metric \(g_\theta \) of nondegenerate CR structure \(({\mathcal {H}},\theta ,J)\) is pseudo-Einstein with Webster scalar curvature \({\hat{r}}\ne 0\), then there exists a real constant \(t\ne 0\) for which the Webster metric associated to \(({\mathcal {H}},t\theta ,J)\) is Einstein–Sasakian.

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References

  1. Abbassi, K.M.T., Kowalski, O.: On Einstein Riemannian g-natural metrics on unit tangent sphere bundles. Ann. Glob. Anal. Geom. 38, 11–20 (2010)

    Article  MathSciNet  Google Scholar 

  2. Abbassi, K.M.T., Calvaruso, G.: g-natural contact metrics on unit tangent sphere bundles. Monatshefte Math. 151, 189–209 (2006)

    MathSciNet  MATH  Google Scholar 

  3. Blair, D.E.: Riemannian Geometry of Contact and Symplectic Manifolds. Progress in Mathematics, vol. 203, 2nd edn. Birkhäuser, Boston (2010)

    Book  Google Scholar 

  4. Boyer, C.P., Galicki, K.: Einstein manifold and contact geometry. Proc. Am. Math. Soc. 129(8), 2419–2430 (2001)

    Article  MathSciNet  Google Scholar 

  5. Cao, H.-D.: Recent Progress on Ricci Solitons, Recent Advances in Geometric Analysis, Advanced Lectures in Mathematics (ALM), vol. 11, pp. 1–38. Int. Press, Somerville (2010)

    MATH  Google Scholar 

  6. Calvaruso, G., Perrone, D.: Contact pseudo metric manifolds. Differ. Geom. Appl. 28, 615–634 (2010)

    Article  MathSciNet  Google Scholar 

  7. Calvaruso, G., Perrone, D.: Erratum to: “Contact pseudo-metric manifolds. Differ. Geom. Appl. 28, 615–634 (2010)”, Differ. Geom. Appl. 31, 836–837 (2013)

  8. Calvaruso, G., Perrone, D.: H-contact semi-Riemannian manifolds. J. Geom. Phys. 71, 11–21 (2013)

    Article  MathSciNet  Google Scholar 

  9. Cho, J.T., Chun, S.H.: Pseudo-Einstein unit tangent sphere bundles. Hiroshima Math. J. 48, 413–427 (2018)

    Article  MathSciNet  Google Scholar 

  10. Cho, J.T.: Pseudo-Einstein manifolds. Topol. Appl. 196, 398–415 (2015)

    Article  MathSciNet  Google Scholar 

  11. Cho, J.T.: Notes on contact Ricci solitons. Proc. Edinb. Math. Soc. 54, 47–53 (2011)

    Article  MathSciNet  Google Scholar 

  12. Cho, J.T.: Strongly pseudo-convex CR space forms. Complex Manifolds 6, 279–293 (2019)

    Article  MathSciNet  Google Scholar 

  13. Chern, S.S., Hamilton, R.S.: On Riemannian Metrics Adapted to Three-Dimensional Contact Manifolds. Lecture Notes in Mathematics, vol. 1111, pp. 279–305. Springer, New York (1985)

    MATH  Google Scholar 

  14. Dragomir, S., Perrone, D.: Harmonic Vector Fields: Variational Principles and Differential Geometry. Elsevier, Amsterdam (2011)

    MATH  Google Scholar 

  15. Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on CR Manifolds, vol. 246. Progress in Mathematics. Birkhäuser, Boston (2006)

    MATH  Google Scholar 

  16. Lee, J.M.: Pseudo-Einstein structures on CR manifolds. Am. J. Math. 110, 157–178 (1988)

    Article  MathSciNet  Google Scholar 

  17. O’Neill, B.: Semi-Riemannian Geometry. Academic Press, New York (1983)

    MATH  Google Scholar 

  18. Perrone, D.: Torsion and critical metrics on contact three-manifolds. Kodai Math. J. 13, 88–100 (1990)

    Article  MathSciNet  Google Scholar 

  19. Perrone, D.: Ricci tensor and spectral rigidity of contact Riemannian 3-manifolds. Bull. Inst. Math. Acad. Sin. 24(2), 127–138 (1996)

    MathSciNet  MATH  Google Scholar 

  20. Perrone, D.: Homogeneous contact Riemannian three-manifolds. Ill. J. Math. 42, 243–256 (1998)

    Article  MathSciNet  Google Scholar 

  21. Perrone, D.: Contact Riemannian manifolds with \(\xi \)-parallel torsion. In: Selected Topics in Cauchy–Riemann Geometry, vol. 9, pp. 307–336. Dip. Mat. Seconda Univ. Napoli, Caserta (2001)

  22. Perrone, D.: Contact metric manifolds whose characteristic vector fied is a harmonic vector field. Differ. Geom. Appl. 20, 367–378 (2004)

    Article  Google Scholar 

  23. Perrone, D.: Geodesic Ricci solitons on unit tangent sphere bundles. Ann. Glob. Anal. Geom. 44(2), 91–103 (2013)

    Article  MathSciNet  Google Scholar 

  24. Perrone, D.: Contact pseudo-metric manifolds of constant curvature and CR geometry. Results Math. 66(1), 213–225 (2014)

    Article  MathSciNet  Google Scholar 

  25. Perrone, D.: Curvature of K-contact semi-Riemannian manifolds. Can. Math. Bull. 57(2), 401–412 (2014)

    Article  MathSciNet  Google Scholar 

  26. Perrone, D.: On the standard nondegenerate almost CR structures of tangent hyperquadrics bundles. Geom. Dedic. 185(1), 15–33 (2016)

    Article  MathSciNet  Google Scholar 

  27. Perrone, D.: Contact semi-Riemannian structures in CR Geometry: some aspects. Appl. Differ. Geom. (Axioms) 8(1), 6 (2019)

    Google Scholar 

  28. Tanaka, N.: A Differential Geometric Study on Strongly Pseudo-Convex Manifolds. Kinokuniya Book Store Co Ltd, Tokyo (1975)

    MATH  Google Scholar 

  29. Tanno, S.: Variational problems on contact Riemannian manifolds. Trans. Am. Math. Soc. 314, 349–379 (1989)

    Article  MathSciNet  Google Scholar 

  30. Tanno, S.: The standard CR structure on the unit tangent bundle. Tôhoku Math. J. 44, 535–543 (1992)

    Article  MathSciNet  Google Scholar 

  31. Webster, S.: Pseudohermitian structures on a real hypersurface. J. Differ. Geom. 13, 25–41 (1978)

    Article  Google Scholar 

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  1. Dipartimento di Matematica e Fisica “E. De Giorgi”, Universitá del Salento, Via Provinciale Lecce-Arnesano, 73100, Lecce, Italy

    Domenico Perrone

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  1. Domenico Perrone
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Correspondence to Domenico Perrone.

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Dedicated to my grandson Davide.

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Perrone, D. On the Pseudohermitian Curvature of Contact Semi-Riemannian Manifolds. Results Math 75, 17 (2020). https://doi.org/10.1007/s00025-019-1137-1

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