$V$-minimal submanifolds

Abstract

We introduce the notion of $V$-minimality, for $V$ a smooth vector field on a Riemannian manifold, a natural extension of the classical notion of minimality, and we prove several basic properties. One featured example is given for locally conformal Kaehler (l.c.K) manifolds. It is well-known that in general, complex submanifolds in non-Kaehler l.c.K manifolds are not minimal. We prove that, however, they are $V$-minimal for $V$ a suitable multiple of the Lee vector field. Extending some results from \cite{AAB}, to emphasis the utility of this notion, we prove that a PHH submersion is $V$-harmonic if and only if it has minimal fibres and a PHH $V$-harmonic submersion pulls back complex submanifolds to $V$ minimal submanifolds.