仿接触几何中的零性条件
Nullity conditions in paracontact geometry
摘要：
本文是一个完整的研究paracontact度量流形的Reeb向量场的基本切触结构满足一个零条件（条件\eqref{paranullity}以下，一些实数$% \tildeκ$和$\tildeμ$）。这类伪黎曼流形，其中包括para-Sasakian流形，最近定义在\cite{MOTE}。本文证明了这些流形与切触度量$(κ,μ)$ -空间之间实际上存在一种对偶。特别是，我们证明，在一些自然的假设下，任何这样的仿接触度量流形承认一个兼容的接触度量$(κ,μ)$ -结构（最终Sasakian）。此外，我们还证明了零性条件在$% \mathcal{D}$ -位似变形下是不变的，并完全确定了整个曲率张量场。最后，在任何维度的非平凡的例子和接触度量的情况下，由于度量的非正定性的许多差异进行了讨论。
英文摘要：
The paper is a complete study of paracontact metric manifolds for which the Reeb vector field of the underlying contact structure satisfies a nullity condition (the condition \eqref{paranullity} below, for some real numbers $% \tildeκ$ and $\tildeμ$). This class of pseudo-Riemannian manifolds, which includes para-Sasakian manifolds, was recently defined in \cite{MOTE}. In this paper we show in fact that there is a kind of duality between those manifolds and contact metric $(κ,μ)$-spaces. In particular, we prove that, under some natural assumption, any such paracontact metric manifold admits a compatible contact metric $(κ,μ)$-structure (eventually Sasakian). Moreover, we prove that the nullity condition is invariant under $% \mathcal{D}$-homothetic deformations and determines the whole curvature tensor field completely. Finally non-trivial examples in any dimension are presented and the many differences with the contact metric case, due to the non-positive definiteness of the metric, are discussed.
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