|This master thesis comprises six chapters.The first chapter consists of introduction and basic notions.The second chapter is based on SO(3) Lie group and so(3) Lie algebra structure. Besides, 1-parameter subgroup of SO(3) and exponential map are defined.In the third chapter, it is given the relation between Riemannian connection and bi-invariant metric. The fourth chapter is formed both SE(3) Lie group structure and se(3) Lie algebra construction. Moreover, Riemannian metric and invariant vector fields on SE(3) are defined. In the fifth chapter, curvatures of left invariant metrics are broughted in. In the last chapter, it is found adjoint representations of so(3) and se(3) Lie algebras as well as shown the relation between bi-invariant metric and adjoint representations.