Curvature tensors in Kaehler manifolds

Author:
Malladi Sitaramayya

Journal:
Trans. Amer. Math. Soc. **183** (1973), 341-353

MSC:
Primary 53B35

DOI:
https://doi.org/10.1090/S0002-9947-1973-0322722-1

MathSciNet review:
0322722

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Curvature tensors of Kaehler type (or type *K*) are defined on a hermitian vector space and it has been proved that the real vector space of curvature tensors of type *K* on *V* is isomorphic with the vector space of sym metric endomorphisms of the symmetric product of , where (Theorem 3.6). Then it is shown that admits a natural orthogonal decomposition (Theorem 5.1) and hence every is expressed as . These components are explicitly determined and then it is observed that is a certain formal tensor introduced by Bochner. We call the *Bochner-Weyl* part of *L* and the space of all these is called the *Weyl subspace* of .

**[1]**Robert C. Gunning and Hugo Rossi,*Analytic functions of several complex variables*, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR**0180696****[2]**Shoshichi Kobayashi and Katsumi Nomizu,*Foundations of differential geometry. Vol I*, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR**0152974****[3]**-,*Foundations of differential geometry*. Vol. II, Interscience Tracts in Pure and Appl. Math., no. 15, Interscience, New York, 1969, MR**38**#6501.**[4]**Alain Lascoux and Marcel Berger,*Variétés Kähleriennes compactes*, Lecture Notes in Mathematics, Vol. 154, Springer-Verlag, Berlin-New York, 1970 (French). MR**0278248****[5]**K. Nomizu,*On the decomposition of generalized curvature tensor fields*(to appear).**[6]**I. M. Singer and J. A. Thorpe,*The curvature of 4-dimensional Einstein spaces*, Global Analysis (Papers in Honor of K. Kodaira), Univ. Tokyo Press, Tokyo, 1969, pp. 355–365. MR**0256303****[7]**K. Yano and S. Bochner,*Curvature and Betti numbers*, Annals of Mathematics Studies, No. 32, Princeton University Press, Princeton, N. J., 1953. MR**0062505**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
53B35

Retrieve articles in all journals with MSC: 53B35

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1973-0322722-1

Keywords:
Hermitian and Kaehler metrics,
curvature tensor,
complex manifold,
hermitian vector space,
fundamental 2-form,
Bochner-Weyl curvature tensor,
Chern class,
Einstein manifold,
tangent bundle

Article copyright:
© Copyright 1973
American Mathematical Society