The structure of supermanifolds
Author:
Marjorie Batchelor
Journal:
Trans. Amer. Math. Soc. 253 (1979), 329338
MSC:
Primary 58A05; Secondary 83E99
MathSciNet review:
536951
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Abstract: The increasing recognition of Lie superalgebras and their importance in physics inspired a search to find an object, a ``supermanifold", which would realize the geometry implicit in Lie superalgebras. This paper analyzes the structure of supermanifolds as defined by B. Kostant. The result is the following structure theorem. The Main Theorem. If E is a real vector bundle over the smooth manifold X, let be the associated exterior bundle and let be the sheaf of sections of . Then every supermanifold over X is isomorphic to for some vector bundle E over X. Although the vector bundle E is not unique but is determined only up to isomorphism, and the isomorphism guaranteed is not canonical, the existence of the isomorphism provides a base for a better understanding of geometry in the graded setting.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197905369510
PII:
S 00029947(1979)05369510
Article copyright:
© Copyright 1979
American Mathematical Society
