2022/05/04 by Vladimir Rovenski, Rovenski, Vladimir
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds
paper · pdf · doi:10.48550/arxiv.2205.02158
openalex publication_date 2022/05/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 f3 + 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.