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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Curvature tensors in Kaehler manifolds

Author: Malladi Sitaramayya
Journal: Trans. Amer. Math. Soc. 183 (1973), 341-353
MSC: Primary 53B35
MathSciNet review: 0322722
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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 $ {\mathcal{L}_K}(V)$ of curvature tensors of type K on V is isomorphic with the vector space of sym metric endomorphisms of the symmetric product of $ {V^ + }$, where $ {V^{\text{C}}} = {V^ + } \oplus {V^ - }$ (Theorem 3.6). Then it is shown that $ {\mathcal{L}_K}(V)$ admits a natural orthogonal decomposition (Theorem 5.1) and hence every $ L \in {\mathcal{L}_K}(V)$ is expressed as $ L = {L_1} + {L_W} + {L_2}$. These components are explicitly determined and then it is observed that $ {L_W}$ is a certain formal tensor introduced by Bochner. We call $ {L_W}$ the Bochner-Weyl part of L and the space of all these $ {L_W}$ is called the Weyl subspace of $ {\mathcal{L}_K}(V)$.

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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