Hostname: page-component-7bb8b95d7b-5mhkq Total loading time: 0 Render date: 2024-09-11T10:48:24.088Z Has data issue: false hasContentIssue false
  • English
  • Français

Volume-preserving geodesic symmetries on four-dimensional Hermitian Einstein spaces

Published online by Cambridge University Press:  22 January 2016

J. T. Cho ,
K. Sekigawa  and
L. Vanhecke
J. T. Cho
Affiliation:
Topology and Geometry, Research Center, Kyungpook National University, Taegu 702-701, Korea
K. Sekigawa
Affiliation:
Department of Mathematics, Niigata University, Niigata 950-21, Japan
L. Vanhecke
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium
Rights & Permissions [Opens in a new window]

Abstract.

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that a four-dimensional Hermitian Einstein space is weakly *-Einsteinian and use this result to show that all geodesic symmetries are volume-preserving (up to sign) if and only if it is local symmetric.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 146 , June 1997 , pp. 13 - 29
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1997

References

[1] Berndt, J., Prüfer, F. and Vanhecke, L., Symmetric-like Riemannian manifolds and geodesic symmetries, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 265–282.Google Scholar
[2] Berndt, J., Tricerri, F. and Vanhecke, L., Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lecture Notes in Math., 1598, Springer-Verlag, Berlin, Heidelberg, New York, 1995.Google Scholar
[3] Besse, A. L., Einstein manifolds, Ergeb. Math. Grenzgeb. 3. Folge 10, Springer-Verlag, Berlin, Heidelberg, New York, 1987.Google Scholar
[4] D’Atri, J. E., Geodesic spheres and symmetries in naturally reductive spaces, Michigan Math. J., 22 (1975), 71–76.Google Scholar
[5] D’Atri, J. E., Nickerson, H. K., Divergence-preserving geodesic symmetries, J. Differential Geom., 3 (1969), 467–476.Google Scholar
[6] D’Atri, J. E., Nickerson, H. K., Geodesic symmetries in spaces with special curvature tensors, J. Differential Geom., 9 (1974), 251–262.Google Scholar
[7] Grantcharov, G. and Muskarov, O., Hermitian *-Einstein surfaces, Proc. Amer. Math. Soc, 120 (1994), 233–239..Google Scholar
[8] Gray, A. and Vanhecke, L., Riemannian geometry as determined by the volumes of small geodesic balls, Acta Math., 142 (1979), 157–198.Google Scholar
[9] Günther, P. and Prüfer, F., D’Atri spaces are ball-homogeneous, preprint 1995.Google Scholar
[10] Kazdan, J. L., personal communication.Google Scholar
[11] Koda, T., A remark on the manifold with Bérard-Bergery’s metric, Ann. Global Anal. Geom., 11 (1993), 323–329.CrossRefGoogle Scholar
[12] Kowalski, O., Spaces with volume-preserving symmetries and related classes of Riemannian manifolds, Rend. Sem. Mat. Univ. Politec Torino, Fascicolo Speciale Settembre (1983), 131–158.Google Scholar
[13] Kowalski, O., Prüfer, F. and Vanhecke, L., D’Atri spaces, Topics in Geometry: In Memory of Joseph D’Atri (Gindikin, S., ed.), Progress in Nonlinear Diff. Eq., 20 (1996), Birkhäuser, Boston, Basel, Berlin, 241–284.CrossRefGoogle Scholar
[14] Brun, C. Le, Einstein metrics on complex surfaces, preprint 1995.Google Scholar
[15] Page, D., A compact rotating gravitational instanton, Phys. Lett., 79B (1979), 235–238.Google Scholar
[16] Prüfer, F., On compact Riemannian manifolds with volume-preserving symmetries, Ann. Global Anal. Geom., 7 (1989), 133–140.Google Scholar
[17] Sekigawa, K., On 4-dimensional connected Einstein spaces satisfying the condition R(X,Y)•R-0, Sem. Rep. Niigata Univ., A7 (1969), 29–31.Google Scholar
[18] Sekigawa, K., On some 4-dimensional compact almost Hermitian manifolds, J. Ramanujan Math. Soc, 2 (1987), 101–116.Google Scholar
[19] Sekigawa, K. and Vanhecke, L., Symplectic geodesic symmetries on Kähler manifolds, Quart. J. Math. Oxford (2), 37 (1986), 95–103.Google Scholar
[20] Sekigawa, K. and Vanhecke, L., Volume-preserving geodesic symmetries on four-dimensional Kähler manifolds, Differential Geometry Peñiscola 1985, Lecture Notes in Math., 1209, Springer-Verlag, Berlin, Heidelberg, New York (1986), pp. 275–291.Google Scholar
[21] Sekigawa, K. and Vanhecke, L., Volume-preserving geodesic symmetries on four-dimensional 2-stein spaces, Kodai Math. J., 9 (1986), 215–224.Google Scholar
[22] Szabó, Z. I., Spectral theory for operator families on Riemannian manifolds, Proc. Sympos. Pure Math., 54 (1993), 615–665.Google Scholar
[23] Tricerri, F. and Vaisman, I., On some 2-dimensional Hermitian manifolds, Math. Z., 192 (1986), 205–216.Google Scholar
[24] Tricerri, F. and Vanhecke, L., Curvature tensors on almost Hermitian manifolds, Trans. Amer. Math. Soc, 267 (1981), 365–398.Google Scholar
[25] Vaisman, I., Non-Kähler metrics on geometric complex surfaces, Rend. Sem. Math. Univ. Politec Torino, 45 (1987), 117–123.Google Scholar
[26] Vanhecke, L., Geometry in normal and tubular neighborhoods, Proc. Workshop on Differential Geometry and Topology Cala Gonone (Sardinia) 1988, Rend. Sem. Fac. Sci. Univ. Cagliari, Supplemento al vol. 58 (1988), 73–176.Google Scholar
[27] Vanhecke, L. and Willmore, T. J., Interaction of tubes and spheres, Math. Ann., 263 (1983), 31–42.Google Scholar
[28] Willmore, T. J., Riemannian geometry, Oxford Science Publications, Clarendon Press, Oxford, 1993.Google Scholar