Curvature tensors in Kaehler manifolds

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 ${\mathcal {L}_K}(V)$.