Classification of totally umbilical $\xi^\perp$ CR-submanifolds of cosymplectic manifolds

Abstract

In \cite{Cab}, Cabras, Ianus and Pitis proved that in a cosymplectic manifold there does not exist any extrinsic sphere tangent to the structure vector field $\xi$. We consider the structure vector field $\xi$ normal to the submanifold in the sense of Papaghiuc \cite{Pap} and derive that a totally umbilical CR-submanifold of a cosymplectic manifold is either (i)~totally geodesic, (ii)~anti-invariant or (iii)~an extrinsic sphere.