Symmetric supermanifolds and Lie triple supersystems

A symmetric supermanifold is defined as a supermanifold $\chi$ endowed with a morphism $\mu$: $\chi\times\chi\to\chi$ satisfying three algebraic properties and a local condition.

To each symmetric supermanifold $\chi$ we associate a Lie triple supersystem, denoted by ${\cal D}$ -($\chi$), and obtain a one-to-one correspondence between simply connected symmetric supermanifolds and Lie triple supersystems. We then show that a symmetric supermanifold is homogeneous. After defining Jordan triples, we show the existence of a distinguished boundary for the symmetric superdomains Gr$\sp\*$ and QGr$\sp\*$.