Invariant Submanifolds of Paracontact Metric $$(\tilde{\kappa }\ne -1, \tilde{\mu })$$-Manifolds

Abstract

The principal objective of this paper is to answer positively the open question whether every invariant submanifold of a paracontact metric $$( \tilde{\kappa },\tilde{\mu })$$ -manifold is totally geodesic. Main result is that any invariant submanifold of a paracontact metric $$(\tilde{\kappa }, \tilde{\mu })$$ -manifold, $$\tilde{\kappa }\ne -1$$ , is always totally geodesic. Additionally, if $$\tilde{\kappa }\ne -1$$ and $$\tilde{\mu }\ne 0$$ the result can be partially reversed, which shows that the totally geodesic submanifold is invariant under the flow of characteristic vector field is tangent to the submanifold.