Mixed Killing vector field and almost coK\"{a}hler manifolds

A vector field $V$ on any (semi-)Riemannian manifold is said to be mixed Killing if for some nonzero smooth function $f$, it satisfies $L_VL_Vg=fL_Vg$, where $L_V$ is the Lie derivative along $V$. This class of vector fields, as a generalization of Killing vector fields, not only identify the isometries of the manifolds, but broadly also contain the class of homothety transformations. We prove an essential curvature identity along those fields on any (semi-)Riemannian manifold and thus generalize the Bochner's theorem for Killing vector fields in this setting. Later we study it in the framework of almost coK\"{a}hler structure and we prove that the Reeb vector field $\xi$ on an almost coK\"{a}hler manifold is mixed Killing if and only if the operator $h=0$. Moving further, we completely classify almost coK\"{a}hler manifolds with $\xi$ mixed Killing vector field in dimension 3. In particular, if $\xi$ on an $\eta$-Einstein almost coK\"{a}hler manifold is mixed Killing, then the manifold is of constant scalar curvature with $h=0$. Also we show that on any $(\kappa,\mu)$-almost coK\"{a}hler manifold, $\xi$ is mixed Killing if and only if the manifold is coK\"{a}hler. In the end we present few model examples in this context.