Lie group structures on smooth manifolds

This study prepared as a graduate thesis covers six chapters. In the first chapter, the historical developments of Lie groups and Lie algebras are expressed and the importance of these topics are discussed. In the second chapter, some basic definitions and theorems on Euclidean spaces and smooth manifolds are given. In the third chapter, some basic facts related to the notions of Riemannian manifolds, connection, Riemannian curvature tensors, sectional curvature, Ricci curvature, scalar curvature are mentioned. In the fourth chapter, some basic facts dealing Lie derivative and integral curve are presented. Some examples of Lie groups are mentioned. Furthermore, the notion of Lie algebra is introduced and Lie algebra of these examples investigated. In the fifth section, some properties of one parameter translation groups, exponential maps and infinitesimal generators defined on Lie groups and Lie algebras are investigated. In the sixth chapter, the theories of Lie groups homomorphisms are presented and the structures of Lie groups are studied. The notions of left invariant, right invariant and bi-invariant metrics are mentioned. Sectional curvature, Ricci curvature and scalar curvature of a Riemannian manifold with a bi-invariant metric are computed.