A Euclidean supermanifold is a supermanifold that can be thought of as being equipped with a flat Riemannian metric.
Alternatively, it is a supermanifold for which the transition functions of an atlas are restricted to be elements of the super Euclidean group.
A Euclidean supermanifold of dimension is a supermanifold that is quipped with an -structure , where and where is the super Euclidean group on .
Here an -structure is defined as follows, essentially being a version of the discussion of pseudogroups at manifold.
Definition (Stolz, Teichner) A -structure on a -dimensional supermanifold consists of
a maximal atlas consisting of charts
(where on the left ) with
such that the transition function
is the restriction of a map
definition A family of -(complex-, super-)manifolds is a map
together with a maximal atlas of charts
such that the transition maps
are the restriction of a map of the following form
example a family of -manifolds, for an ordinary manifold is a submersion with flat Riemannian metric on the fibers.
ordinary Euclidean manifolds as Euclidean supermanifolds
Specifically in 2-dimensions, an ordinary Spin-Eulidean manifold is one with -structure.
We want to regard this as a Euclidean supermanifold with -structure.
In general for two structures and we can transfer structures when we have a group homomorphisms
and with respect to that a
-equivariant map .
Then send every -chart to the corresponding -chart which as a subset of is the inverse image of .
This yields a functor