Submanifolds of framed metric manifolds and $S$-manifolds

Abstract

Almost semi-invariant submanifolds of framed metric manifolds and $S$-manifolds are studied, where the ambient manifolds generalize almostHermitian, Kaehler, almost contact metric and Sasakian manifolds while thesubmanifolds generalize/imply several known classes of submanifoldsincluding $CR$, semi-invariant, holomorphic, totally real and slantsubmanifolds. Integrability conditions for certain natural distributions onalmost semi-invariant submanifolds of $S$-manifolds are investigated.Parallelism of the operators arising naturally in the study is investigatedleading to a flowchart connecting them. Totally umbilical, totally geodesicand totally contact geodesic submanifolds are studied. Some relationsbetween almost semi-invariant submanifolds of a framed metric manifold andother submanifolds are investigated. Submanifolds of principal toroidalbundles are also studied.