Section 4 Analytic Vector Fields

Abstract

Publisher SummaryThis chapter describes the importance of analytic vector fields. Tangent vectors and vector fields are basic algebraic objects associated with a Banach manifold. It is found that of particular importance is the commutator product for analytic vector fields, giving rise to a Lie algebra structure. Many deep geometric and analytic properties of certain Banach manifolds are obtained by studying Lie algebras of analytic vector fields. As in the finite-dimensional case, the tangent vectors for Banach manifolds can be defined using the concept of germ of an analytic mapping. It is observed that as in the finite-dimensional case, tangent vectors can also be defined as directional derivatives along smooth curves. The local representation of tangent vectors in terms of a chart carries over to analytic vector fields. It is found for several important classes of complex Banach manifolds that every analytic vector field is actually a polynomial vector field with respect to a suitable canonical chart, and is therefore algebraic in character. The analytic automorphisms of complex manifolds are often given by algebraic expressions.