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Zbl 0874.53024
Bonome, Agustín; Castro, Regina; García-Río, Eduardo; Hervella, Luis M.; Matsushita, Y.
Null holomorphically flat indefinite almost Hermitian manifolds.
(English)
[J] Ill. J. Math. 39, No.4, 635-660 (1995). ISSN 0019-2082

Let $(M,g,J)$ be an indefinite almost Hermitian manifold, that is, $(M,g)$ is a semi-Riemannian manifold with indefinite metric $g$ and $J$ is an almost complex structure on $M$ such that $g(Ju,Jv)=g(u,v)$, for all tangent vectors $u$, $v$ on $M$. In this paper, the authors systematically study the behaviour of the curvature tensor $R$ of $(M,g)$ on holomorphic (i.e., $J$-invariant) tangent planes $\pi=\text{span}\{v,Jv\}$. Such a plane is degenerate if and only if $v$ is a null vector, which means $g(v,v)=0$, $(v\ne 0)$. An indefinite almost Hermitian manifold $(M,g,J)$ is said to be null holomorphically flat if $g(R(v,Jv)v,Jv)=0$ for all null tangent vectors $v$. Several nice examples are shown to point out that there exist null holomorphically flat almost Hermitian manifolds which are not of constant holomorphic sectional curvature. It is proved that an indefinite almost Hermitian manifold $(M,g,J)$ is null holomorphically flat if and only if it satisfies $R(u,Ju)Ju+JR(u,Ju)u=c(u)u$ for all null tangent vectors $u$. The function $c(u)$ in this result can be seen as a measure of the failure of a null holomorphically flat almost Hermitian manifold to have pointwise constant holomorphic sectional curvature.\par It is also shown the following relation between bounds on the holomorphic sectional curvature and the sign of $c(u)$: (1) $(M,g,J)$ is null holomorphically flat and $c(u)\leq 0$ if and only if the holomorphic sectional curvature is bounded from below on spacelike planes and from above on timelike planes, and (2) $(M,g,J)$ is null holomorphically flat and $c(u)\geq 0$ if and only if the holomorphic sectional curvature is bounded from above on spacelike planes and from below on timelike planes.\par By using the (usual) Ricci tensor $\rho$ and the $*$-Ricci tensor $\rho^*$ of an indefinite almost Hermitian manifold $(M,g,J)$, it is obtained that if $(M,g,J)$ is null holomorphically flat, then $c(u)=(-1/(2n+4))\{\rho(u,u)+\rho(Ju,Ju)+6\rho^*(u,u)\}$, where $2n=\dim(M)$, for all null tangent vectors $u$ on $M$. This result plays a fundamental role later in the explicit determination of the curvature tensor of a null holomorphically flat indefinite almost Hermitian manifold, such that the known expression of the curvature of an almost Hermitian manifold of pointwise constant holomorphic sectional curvature is generalized.\par Then, a natural criterion for a null holomorphically flat manifold to have pointwise constant holomorphic sectional curvature is stated. Finally, some local decomposition theorems for null holomorphically flat indefinite almost Hermitian manifolds under extra geometric assumptions are given.
MSC 2000:
*53C15 Geometric structures on manifolds
53C55 Complex differential geometry (global)
53B30 Lorentz metrics, indefinite metrics
53C50 Lorentz manifolds, manifolds with indefinite metrics

Keywords: indefinite almost Hermitian manifold; constant holomorphic sectional curvature; null holomorphically flat; decomposition theorems

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