Generalized CoK\"ahler Geometry and an Application to Generalized K\"ahler Structures

In this paper we define the notion of a generalized coK\"ahler structure and prove that the product $M_{1}\times M_{2}$ of generalized contact metric manifolds $(M_i, \Phi_i,E_{\pm,i}, G_i)$, $i=1, 2$, where $M_{1}\times M_{2}$ is endowed with the product generalized complex structure induced from $\Phi_1$ and $\Phi_2$, is generalized K\"ahler if and only if $(M_i, \Phi_i, E_{\pm,i}, G_i) ,\ \ i=1,2$ are generalized coK\"ahler structures. We also prove that products of generalized coK\"ahler and generalized K\"ahler manifolds admit a generalized coK\"ahler structure. We use these product constructions to give nontrivial examples of generalized coK\"ahler structures. Finally, we show the analogs of these theorems hold in the setting of twisted generalized geometries. We use these theorems to construct new examples of twisted generalized K\"ahler structures on manifolds that do not admit a classical K\"ahler structure and we give examples of twisted generalized coK\"ahler structures on manifolds which do not admit a classical coK\"ahler structure.