Manifolds with generalized boundary and differentiable semigroups

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1979

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Abstract

This work is devoted to the development of a differential calculus on manifolds with generalized boundary and an investigation of differentiability in semigroups on such spaces. A manifold with generalized boundary is a second countable Hausdorff space such that each point has an open neighborhood homeomorphic to a subset of Euclidean space with dense interior. First, the notion of a Frechet derivative is strengthened to apply to vector-valued functions on subsets of Euclidean space with dense interior, and the possibility of extending differentiable functions is examined for certain special cases. The chain rule, inverse function theorem, and several other theorems from advanced calculus are established in this context. The differential geometry of manifolds with generalized boundary is developed next. Attention is given to differentiable structures, the tangent bundle, products, vector fields, and the Lie bracket operation on vector fields. Finally, differentiable semigroups are investigated. A differentiable semigroup is a semigroup S on an n-manifold with generalized boundary such that the multiplication function is differentiable. It is shown that if S has an identity, then the multiplication satisfies a cancellation law in a neighborhood of the identity. The vector space of all left-invariant vector fields of S is shown to be a Lie algebra of dimension n when S has an identity and the multiplication of S is three times differentiable. If the multiplication of S is infinitely differentiable and S has an identity, then S is locally embeddable in a Lie group and if, in addition, S has a smooth boundary and the identity is on the boundary, then the group of units of S is an open subset of the boundary.

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