ON THE STRUCTURE OF HOMOGENEOUS SUPERMANIFOLDS

Let $(G,\mathcal O_G)$ be a real or complex Lie supergroup and let$(H,\mathcal O_H)$ be a Lie subsupergroup. We give an explicitconstruction of a supermanifold structure on the coset space $G/H$.The construction uses a local transversal subsupermanifold to$(H,\mathcal O_H)$ and its translates. It yields a smoothsupermanifold in the real case and a holomorphic supermanifold inthe complex case. We describe the resulting structure sheaf interms of right $(H,\mathcal O_H)$-invariant superfunctions and showthat the natural action of $(G,\mathcal O_G)$ on$(G/H,\mathcal O_{G/H})$ is transitive. Conversely, we prove thatevery real or complex homogeneous supermanifold is equivariantlyisomorphic to a coset supermanifold of this form.