We analyze the category of GH^\infty supermanifolds recently introduced by Rogers and show that these supermanifolds do not have a good graded tangent bundle, and that a natural definition of super vector bundle is not possible within that category. However, any GH^\infty supermanifold can be turned into a supermanifold of a new category (that we call a G-supermanifold) which is well-behaved and is a particular case of a supermanifold à la Rothstein.
Some remarks on the differential-geometric approach to supermanifolds
BARTOCCI, CLAUDIO;
1987-01-01
Abstract
We analyze the category of GH^\infty supermanifolds recently introduced by Rogers and show that these supermanifolds do not have a good graded tangent bundle, and that a natural definition of super vector bundle is not possible within that category. However, any GH^\infty supermanifold can be turned into a supermanifold of a new category (that we call a G-supermanifold) which is well-behaved and is a particular case of a supermanifold à la Rothstein.
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