论文标题
关于削弱的$ f $结构的几何形状
On the geometry of a weakened $f$-structure
论文作者
论文摘要
K. Yano于1963年推出的$ f $ - 结构，随后由许多几何图形进行了研究，是一个几乎复杂且几乎接触结构的更高维度的类似物，由A（1,1）-tensor field $ f $ on A $（2n + P）上的$ f $ f $ - $（2n + p）$ - dimensional-dimensiation-dimensiation-dimensiation-dimensy Formold，它满足$ f^3 + f = 0 $ $ $ 2n $ 2n $ 2n $ 2n $ 2n $ 2n。 We recently introduced the weakened (globally framed) $f$-structure (i.e., the complex structure on $f(TM)$ is replaced by a nonsingular skew-symmetric tensor) and its subclasses of weak $K$-, ${\cal S}$-, and ${\cal C}$- structures on Riemannian manifolds with totally geodesic foliations, which allow us to重新看一下古典理论。我们通过对全球框架$ f $ - manifolds的几个已知结果概括来证明这一点。首先，我们使用新的张量$ f $ f $ - 结构来表达$ f $的协方差衍生物，然后我们证明，在弱$ k $ - manifold上，特征矢量场正在杀死，$ \ ker f $定义了一个完全地理的粮食，$ {$ {\ cal s} $ - 结构$ - 结构; $ {\ cal s} $ - 结构，以及带有并行张量$ f $的度量弱$ f $ - 结构，将其减少到弱$ {\ cal c} $ - 结构。对于$ p = 1 $，我们获得了相应的定义，用于弱接触，弱余弦和弱的Sasakian结构。
An $f$-structure, introduced by K. Yano in 1963 and subsequently studied by a number of geometers, is a higher dimensional analog of almost complex and almost contact structures, defined by a (1,1)-tensor field $f$ on a $(2n+p)$-dimensional manifold, which satisfies $f^3 + f = 0$ and has constant rank $2n$. We recently introduced the weakened (globally framed) $f$-structure (i.e., the complex structure on $f(TM)$ is replaced by a nonsingular skew-symmetric tensor) and its subclasses of weak $K$-, ${\cal S}$-, and ${\cal C}$- structures on Riemannian manifolds with totally geodesic foliations, which allow us to take a fresh look at the classical theory. We demonstrate this by generalizing several known results on globally framed $f$-manifolds. First, we express the covariant derivative of $f$ using a new tensor on a metric weak $f$-structure, then we prove that on a weak $K$-manifold the characteristic vector fields are Killing and $\ker f$ defines a totally geodesic foliation, an ${\cal S}$-structure is rigid, i.e., our weak ${\cal S}$-structure is an ${\cal S}$-structure, and a metric weak $f$-structure with parallel tensor $f$ reduces to a weak ${\cal C}$-structure. For $p=1$ we obtain the corresponding corollaries for weak almost contact, weak cosymplectic, and weak Sasakian structures.
