Geometric properties of involutive distributions on graded manifolds

Lie supergroups supported over and associated to the adjoint representation

“…The point is that the introduction of the odd generators of the Lie superalgebra does not alter this fact. This has been discussed and elucidated in Theorem 6 and Corollary 9 of [7].…”

“…Even though this remark is very well understood in the classical Lie theory, we now want to see how the quoted results from [7] get realized when we include the contributions coming from the integral flows of the odd vector fields representing the odd Lie algebra generators y 0 , y 1 , y 2 and y 3 . As mentioned before, the techniques introduced in [6] can be readily applied and in this case, the integral flow Γ y i : R 1|1 × F 2|2 → F 2|2 depends on an odd parameter τ i , as (cf.…”

Secondary characteristic classes of super-foliations

“…Proof. Since the 1-form a defines a super-foliation of codimension 0 + 1, there exists an odd 1-form b such that da = b ∧ a, see [8]. Since b is odd, we have b ∧ b = 0 and the identity 0…”

Foliations on supermanifolds and characteristic classes

“…Definition of a superfoliation. We recall the definition of a superfoliation of codimension n + εm [Leites 1980;Monterde et al 1997;Tuynman 2004]. First, a distribution of codimension n+εm is a sub-supervector bundle of T ᏹ of dimension…”

“…Theorem 2.10 [Hill and Simanca 1991;Monterde et al 1997]. Any superfoliation of codimension n + εm on a supermanifold of dimension p + εq is locally isomorphic to the elementary superfoliation ‫ޒ‬ p,q n,m .…”