On the splitting of weak nearly ${\cal C}$-manifolds
Sourav Nayak, Dhriti Sundar Patra, Vladimir Rovenski
TL;DR
The paper analyzes weak nearly ${\cal C}$-manifolds, a generalization of nearly ${\cal C}$-manifolds in the framework of weak metric $f$-structures, to determine when such manifolds split locally as Riemannian products. It introduces and studies the family of Reeb-type tensors $h_i=\nabla\xi_i$, proves their commutativity and spectral structure, and leverages these to establish a splitting theorem for $(2n+s)$-dimensional manifolds under mild additional conditions, with a complete characterization in the $(4+s)$-dimensional case. When the combined obstruction ${\bf h}$ vanishes, the manifold decomposes as $\mathbb{R}^s\times\bar{M}^{2n}$ with $\bar{M}$ carrying a weak nearly Kähler structure, while in the general case a product $B^{4+s}\times\bar{M}^{2n-4}$ arises with $\bar{M}$ weak nearly Kähler and $B$ satisfying a related weak nearly ${\cal C}$-structure. These results extend rov-128 and NDY-2018 to multi-Reeb settings and yield new insights for nearly ${\cal C}$-manifolds, including curvature-invariance and integrability of constructed leaves, with potential applications to the study of $\mathfrak{g}$-foliations and $s$-cosymplectic structures.
Abstract
The interest of mathematicians in metric $f$-manifolds, in particular, almost contact metric manifolds, is motivated by the study of the geometry and dynamics of contact foliations, as well as their applications in physics. Weak metric $f$-manifolds, defined by V. Rovenski and R. Wolak (2022), open a new perspective on classical theory of $f$-manifolds and discover new applications. In this paper, we study manifolds of this type, called weak nearly ${\cal C}$-manifolds, which generalize almost ${\cal C}$-manifolds. We find conditions under which a $(2n+s)$-dimensional weak nearly ${\cal C}$-manifold becomes locally a Riemannian product, and characterize $(4+s)$-dimensional weak nearly ${\cal C}$-manifolds. The consequences of these theorems present new results for nearly ${\cal C}$-manifolds.