Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Almost hermitian manifolds and Osserman condition

  • 40 Accesses

  • 2 Citations

Abstract

Let(M, g, J) be an almost Hermitian manifold. In this paper we study holomorphically nonnegatively,Δ)-pinched almost Hermitian manifolds. In [3] it was shown that for such Kahler manifolds a plane with maximal sectional curvature has to be a holomorphic plane(J-invariant). Here we generalize this result to arbitrary almost Hermitian manifolds with respect to the holomorphic curvature tensorH R and toRK-manifolds of a constant type λ(p). In the proof some estimates of the sectional curvature are established. The results obtained are used to characterize almost Hermitian manifolds of constant holomorphic sectional curvature (with respect to holomorphic and Riemannian curvature tensor) in terms of the eigenvalues of the Jacobi-type operators, i.e. to establish partial cases of the Osserman conjecture. Some examples are studied.

This is a preview of subscription content, log in to check access.

References

  1. [1]

    V. Apostolov, J. Davidov, andO. Muškarov, Compact self-dual Hermitian surfaces,Trans. of AMS,348,8 (1996), 3051–3063.

  2. [2]

    H. Baum, T. Friedrich, R. Grunewald, andI. Kath,Twistor and Killing spinors on Riemannian manifolds, Seminarbericht Nr.108, Humboldt-Universität, Berlin, 1990.

  3. [3]

    B. Bishop andS. Goldberg, On the topology of positively curved Kähler manifolds,Tôhoku Math. J. 15 (1993), 359–364.

  4. [4]

    Q.S. Chi, A curvature characterization of certain locally rank-one symmetric spaces,J. Differential Geom. 28 (1988) 187–202.

  5. [5]

    J. Davidov, O. Muškarov andG. Grantcharov, Kähler curvature identities for twistor spaces.Illinois J. of Math. 39 No. 4, (1995), 627–634.

  6. [6]

    A. Derdziński, Self-dual Kähler manifolds and Einstein manifolds of dimension four,Compositio Math. 49 (1983), 405–433.

  7. [7]

    G. Ganchev, On Bochner curvature tensor in almost Hermitian manifolds, Pliska,Studia mathematica Bulgarica 9 (1987), 33–43.

  8. [8]

    P. Gilkey, Manifolds whose curvature operator has constant eigenvalues at the base-point,J. Geom. Anal. 4 (1994), 155–158.

  9. [9]

    P. Gilkey, A. SWAN, andL. Vanhecke, Isoparametric geodesic spheres and a conjecture of Osserman concerning the Jacobi operator,Quart. J. Math. Oxford. 46 (1995), 299–320.

  10. [10]

    A. Gray, Riemannian manifolds with geodesic symmetries of order 3,J. Differential Geometry 7 (1972), 343–369.

  11. [11]

    —, Nearly Kähler manifolds,J. Differential Geometry 4 (1970), 283–309.

  12. [12]

    —, Classification des variétés approximativement kählériennes de courbure sectionelle holomorphe constante,C. R. Acad. Sc. Paris, Série A,279 (1974), 797–800.

  13. [13]

    A. Gray andL. Vanhecke, Almost Hermitian manifolds with constant holomorphic sectional curvature,Časopis pro pěstováni matematiky, Praha 104 (1979), 170–179.

  14. [14]

    Z. Olszak, On the existence of generalized complex space forms,Israel J. Math. 65 (1989), 214–218.

  15. [15]

    R. Osserman, Curvature in the eighties,Amer. Math. Monthly 97 (1990), 731–756.

  16. [16]

    M. Prvanović, On the curvature tensor of Kähler type in an almost Hermitian and almost para-Hermitian manifold,Matematički vesnik 50 (1998), 57–64.

  17. [17]

    K. Sekigawa and L. Vanhecke, Volume preserving geodesic symmetries on four dimensional Kähler manifolds,Differential geometry Penścola, 1985, Proceedings (A. M. Naveira, A. Fernandez, and F. Mascaro, eds.), Lecture Notes in Math.1209, Springer, 275–290.

  18. [18]

    F. Tricerri andL. Vanhecke, Curvature tensors of almost Hermitian manifolds,Trans, of AMS 267 (1981), 365–398.

  19. [19]

    L. Vanhecke, Almost Hermitian manifolds withJ-invariant Riemannian curvature tensor,Rend. Sem. Math. Univers. Politecn. Torino 34 (1975–76), 487–497.

  20. [20]

    —, The Bochner curvature tensor on almost Hermitian manifolds,Rend. Sem. Math. Univers. Politecn. Torino 34 (1975–76), 21–37.

  21. [21]

    —, Some almost Hermitian manifolds with constant holomorphic sectional curvature,J. Diff. Geometry 12 (1977), 461–471.

  22. [22]

    L. Vanhecke andF. Bouten, Constant type for almost Hermitian manifolds,Bull. Math, de la Soc. Set. Math, de la R.S. Roumanie 20(68),3–4 (1976), 415–422.

Download references

Author information

Correspondence to N. Blažić or M. Prvanović.

Additional information

The first author is partially supported by SFS, Project #04M03.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Blažić, N., Prvanović, M. Almost hermitian manifolds and Osserman condition. Abh.Math.Semin.Univ.Hambg. 71, 35–47 (2001). https://doi.org/10.1007/BF02941459

Download citation

Keywords

  • Curvature Tensor
  • Twistor Space
  • Killing Spinor
  • Jacobi Operator
  • Hermitian Manifold