On submanifolds of a cosymplectic manifold (Q2748620)

The authors study CR submanifolds of a cosymplectic manifold, i.e. submanifolds of an almost contact metric manifold with parallel structural one and two forms, which admit a Cauchy-Riemann decomposition of their tangent bundle into an invariant and a real sub-bundle. NEWLINENEWLINENEWLINEFirst, a classification of such submanifolds with totally umbilical real sub-bundle is given. Then, for CR submanifolds of a cosymplectic space form with structure vector field in their invariant distribution, a global upper bound of the \(\varphi\)-sectional curvature is given. Equality is shown to occur if and only if the invariant distribution is totally geodesic. NEWLINENEWLINENEWLINEFinally, for semi-invariant \(\xi^{\perp}\)-submanifolds of a cosymplectic manifold, i.e. submanifolds with a decomposition of their tangent space into invariant and real sub-bundles, a necessary and sufficient condition is given for the integrability of the invariant distribution, while its orthogonal complement is shown to be always integrable. An example of such a submanifold is given.