Mathematische Zeitschrift

Let S be a compact Riemann surface and let H be a finite group. It is known that if H acts on S,  then there is a H-equivariant isogeny decomposition of the Jacobian variety JS of S,  called the group algebra decomposition of JS with respect to H. If $$S_1 \rightarrow S_2$$ is a regular covering map, then it is also known that the group algebra decomposition of $$JS_1$$ induces an isogeny decomposition of $$JS_2.$$ In this article we deal with the converse situation. More precisely, we prove that the group algebra decomposition can be lifted under regular covering maps, under appropriate conditions.

It is well known that a Hopf vector field on the unit sphere S 2n+1 is the Reeb vector field of a natural Sasakian structure on S 2n+1. A contact metric manifold whose Reeb vector field ξ is a harmonic vector field is called an H-contact manifold. Sasakian and K-contact manifolds, generalized (k, μ)-spaces and contact metric three-manifolds with ξ strongly normal, are H-contact manifolds. In this paper we study, in dimension three, the stability with respect to the energy of the Reeb vector field ξ for such special classes of H-contact manifolds (and with respect to the volume when ξ is also minimal) in terms of Webster scalar curvature. Finally, we extend for the Reeb vector field of a compact K-contact (2n+1)-manifold the obtained results for the Hopf vector fields to minimize the energy functional with mean curvature correction.