Exponential Maps: Properties and Applications in Lie Groups

This lecture discusses the properties of the exponential map in the context of Lie groups and their algebras. The instructor begins by defining the exponential map and its significance in connecting the Lie algebra to the Lie group. The discussion includes the smoothness of the exponential map, emphasizing that it is a local diffeomorphism at the origin. The instructor illustrates how integral curves relate to the exponential map and demonstrates the extension of these curves over larger intervals. The lecture also covers the bijection between connected Lie subgroups and their corresponding Lie algebras, highlighting the implications of this relationship. The instructor provides examples and proofs to clarify these concepts, including the behavior of the exponential map under group morphisms. The session concludes with a discussion on the implications of isomorphic Lie algebras and their corresponding Lie groups, setting the stage for further exploration of the topology of Lie groups and their universal covers.