Topics In The Differential Geometry of Supermanifolds

The book combines the algebraic and differential
geometric approaches to supermanifold theory. It
begins with a superspace, constructed by taking the
Cartesian product of copies of the even and odd parts
of an infinite-dimensional Banach Grassmann algebra.
The superspace serves as a model space of G-infinite
supermanifolds. The theory of super Lie groups and
super Lie algebras, (in particular the existence
theorems on super Lie groups) is given. Super
principal fiber bundles equipped with connections and
the action of super Lie groups and supervector fields
are introduced. The concept of parallel transport
along smooth curves is introduced in the same way as
in conventional differential geometry. It is shown
that a smooth curve in the base space can be lifted
uniquely into the bundle and that parallel
displacements along closed smooth curves form a
group, the super holonomy group . It is further
shown that the group is a sub-super Lie group of the
structure group of the bundle. The result follows
from the super analogues of the Freudenthal theorem
and the reduction theorem of bundles. The book
concludes with a proof of a super version of the
Ambrose-Singer holonomy theorem.

Autorentext
Education: Univ. of Cape Coast, 1977- 81; BS Math, Dip. Ed. University of Ghana 1989-91; MPhil Math Physics. Howard Univ 1992-2002; Ph.D. GWU 2006 -08; Grad Cert. Survey Design. Employment: Assoc. Professor of Math Howard Community College 1995-present. Honoree in "Who's Who Among America's Teachers" in 2003-04, 2004-05, and 2005-06.

Klappentext
The book combines the algebraic and differential geometric approaches to supermanifold theory. It begins with a superspace, constructed by taking the Cartesian product of copies of the even and odd parts of an infinite-dimensional Banach Grassmann algebra. The superspace serves as a model space of G-infinite supermanifolds. The theory of super Lie groups and super Lie algebras, (in particular the existence theorems on super Lie groups) is given. Super principal fiber bundles equipped with connections and the action of super Lie groups and supervector fields are introduced. The concept of parallel transport along smooth curves is introduced in the same way as in conventional differential geometry. It is shown that a smooth curve in the base space can be lifted uniquely into the bundle and that parallel displacements along closed smooth curves form a group, the "super holonomy group". It is further shown that the group is a sub-super Lie group of the structure group of the bundle. The result follows from the super analogues of the Freudenthal theorem and the reduction theorem of bundles. The book concludes with a proof of a super version of the Ambrose-Singer holonomy theorem.