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.