Abstract
In this chapter we treat the theory of principal bundles and connections on them for families of supermanifolds. The most important examples of principal bundles are frame bundles of vector bundles. Many extra structures on vector bundles, such as metrics or almost complex structures can actually be formulated in terms of a reduction of the structure group of the frame bundle of the vector bundle. The theory of connections on principal bundles sheds light on properties of covariant derivatives that are compatible with such extra structures.
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