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On higher covariant derivatives of the curvature tensors of Kählerian C-spaces

Published online by Cambridge University Press:  22 January 2016

Ryoichi Takagi*
Affiliation:
Department of Mathematics, University of Tsiikuba, Sakura-mura, Niihari-gun, Ibaraki, Japan
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A compact simply connected complex homogeneous manifold is said briefly a C-space, which was completely classified by H. C. Wang [12]. A C-space is called to be Kählerian if it admits a Kählerian metric such that a group of isometries acts transitively on it. Hermitian symmetric spaces of compact type are typical examples of a Kählerian C-space. Let M be an arbitrary Kählerian C-space and R its curvature tensor. M. Itoh [6] expressed R in the language of Lie algebra and investigated various properties of R. In this paper, we study higher covariant derivatives of R.

Type
Research Article
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1983

References

[1] Borel, A., Kâhlerian coset spaces of semi-simple Lie groups, Proc. Nat. Acad. Sci.U. S. A., 40 (1954), 11471151.CrossRefGoogle Scholar
[2] Borel, A. and Hirzebruch, F., Characteristic classes and homogeneous spaces I, Amer. J. Math., 80 (1958), 458-538.Google Scholar
[3] Bourbaki, N., Groups et algebres de Lie IV, V et VI, Hermann, Paris, 1968.Google Scholar
[4] Cartan, E., Les groupes reels simples, finis et continus, Œuvres Completes, Partie I, Volume 1 (1952), 399491.Google Scholar
[5] Freudenthal, Hans and Vries, H. de, Linear Lie Groups, Academic Press, 1969.Google Scholar
[6] Itoh, M., On curvature properties of Kähler C-spaces, J. Math. Soc. Japan, 30(1976), 3971.Google Scholar
[7] Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry II, Interscience Publishers, 1969.Google Scholar
[8] Nakagawa, H. and Takagi, R., On locally symmetric Kaehler submanifolds in a complex projective space, J. Math. Japan, 28 (1976), 638667.Google Scholar
[9] Nomizu, K., Invariant affine connections on homogeneous spaces, Amer. J. Math., 76 (1954), 33-65.Google Scholar
[10] Takagi, R., Kahlerian submanifolds in a complex projective space with second fundamental form of polynomial type, to appear.Google Scholar
[11] Takeuchi, M., Homogeneous Kähler submanifolds in complex projective spaces, Japan. J. Math., 4 (1978), 171219.Google Scholar
[12] Wang, H. C., Closed manifolds with homogeneous complex structures, Amer. J. Math., 76 (1954), 132.CrossRefGoogle Scholar
[13] Wolf, J. A., On the classification of hermitian symmetric spaces, J. Math. Mech., 13 (1964), 489495.Google Scholar