Characterization of Sasakian Manifolds by Curvature Conditions

Sasakian manifolds, a subclass of contact metric manifolds, play a central role in modern differential geometry and mathematical physics, where they arise naturally in geometric mechanics, CR geometry, and supersymmetric field theories. The weak contact metric (w.c.m.) structure generalizes the contact metric structure and provides a broader framework for studying contact geometry and its applications. A fundamental problem is to understand how curvature conditions constrain the underlying contact metric structure and, in particular, how they distinguish Sasakian geometry from its weaker variants. Using the partial Ricci flow, we characterize Sasakian structure (among w.c.m. manifolds) under the $(\kappa ,\mu )$-nullity condition related to curvature. For $\kappa <1$ we find conditions under which a w.c.m. manifold admits a bi-Legendrian structure, and for $\kappa =\mu =0$ establish splitting and classification results.