Reduction of coK\"{a}hler and 3-cosymplectic manifolds

The notions of coK\"{a}hler manifolds and 3-cosymplectic manifolds are odd-dimensional analogues of the ones of K\"{a}hler manifolds and hyperK\"{a}hler manifolds, respectively. In this paper, we obtain reduction theorems of coK\"{a}hler manifolds and 3-cosymplectic manifolds. We show that K\"{a}hler and coK\"{a}hler (hyperK\"{a}hler and 3-cosymplectic) reductions admit a natural compatibility with respect to ``cylinder constructions". We further prove the compatibility of K\"{a}hler and coK\"{a}hler reductions with respect to the "mapping torus construction".