Curvature Tensors on Almost Hermitian Manifolds

Abstract

Abstract.A complete decomposition of the space of curvature tensors over a Hermitian vector space into irreducible factors under the action of the unitary group is given. The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for almost Hermitian manifolds are given. Conformal invariants are considered and a general Bochner curvature tensor is introduced and shown to be a conformal invariant. Finally curvature tensors on four-dimensional manifolds are studied in detail.1. Introduction. Let (V, g) be an n-dimensional real vector space with positive definite inner product g and denote by 61 (V) the subspace of V* <8> V* <8> V* <8> V* consisting of all tensors having the same symmetries as the curvature tensor of a Riemannian manifold, including the first Bianchi identity. In a well-known paper [21] Singer and Thorpe considered 61(F) (in particular for n = 4) and gave a geometrical useful description of the splitting of 61(F) under the action of &(n) into three components. This was also studied by Nomizu [18] for generalized curvature tensor fields.A similar decomposition was given in [16], [17] and [22] when V is a 2«-dimensional real vector space endowed with a complex structure J compatible with a positive definite inner product g and for the subspace %(V) of 61(F) consisting of

Local models and integrability of certain almost Kähler 4-manifolds

“…In the presence of a g-orthogonal almost complex structure J , which induces the chosen orientation of M, the above SO(4)-decomposition can be further refined to get seven irreducible U(2)-invariant pieces [47]. To see this, we first decompose the traceless Ricci tensor into its J -invariant and J -anti-invariant parts, Ric 0 and Ric 0 = Ric , which gives rise to the decomposition Ric 0 = Ric 0 Ric 0 .…”

Geometric Realizations of Affine Kähler Curvature Models

“…For example, the proof of Theorem 1.2 rests upon work of Higa [9]. The proof of Theorem 1.5 and the proof of Theorem 1.6 rest upon the decomposition of Tricerri and Vanhecke [14]. The proof of Theorem 1.8 will rest upon the decomposition of K(V, J) as a unitary module given in [10][11][12].…”

“…A(e 1 , f 1 )e 2 = −2 2 e 2 , ρ 14 Next take Θ 111 = 1 (x 2 + √ −1y 2 ) and Θ 222 = 2 (x 1 + √ −1y 1 ). Again, let A := R(0).…”

“…These tensors are anti-symmetric in the last two indices so ρ 13 = −ρ 14 . We verify that 0 = A 1 − A 2 ∈ W 9 ∩ J + which establishes Assertion (2).…”

Conformally invariant tensors of an almost Hermitian manifold associated with the holomorphic curvature tensor

“…Thus, tensor B(HR), defined by(3.4), is called generalized Bochner curvature tensor.In[2,5] the tensor defined by (3.4) is obtained applying the technics of decomposition of the Hermitian vector space under the action of the unitary group. Here, it is obtained by a direct application of the conformal change of the metric.…”