An introduction to the geometrical analysis of vector fields : with applications to maximum principles and lie groups

Description

This book provides the reader with a gentle path through the multifaceted theory of vector fields, starting from the definitions and the basic properties of vector fields and flows, and ending with some of their countless applications, in the framework of what is nowadays called Geometrical Analysis. Once the background material is established, the applications mainly deal with the following meaningful settings: Flows of vector fields in space -- The exponential theorem -- The composition of flows of vector fields -- Hadamard's theorem for flows -- The CBHD operation on finite dimensional lie algebras -- The connectivity theorem -- The Carnot-Carathéodory distance -- The weak maximum principle -- Corollaries of the weak maximum principle -- The maximum propagation principle -- The maximum propagation along the drift -- The differential of the flow wrt its parameters -- The exponential theorem for ODEs -- The exponential theorem for lie groups -- The local third theorem of lie -- Construction of carnot groups -- Exponentiation of vector field algebras into lie groups -- On the convergence of the CBHD series