ABSTRACT
This chapter introduces a Riemannian metric by generalizing a procedure used by V. I. Arnold to introduce an invariant Riemannian metric on a Lie group while studying the hydrodynamics of perfect fluids. A Riemannian metric on the Lie group will be called left-invariant if it is preserved by all left translations i.e. if the induced linear map of a left translation carries every vector to a vector of the same length. A Riemannian metric is called right-invariant if it is invariant under right translations. The chapter presents some generalities on principal fiber bundles and their associated vector bundles. It also provides a discussion on the definition of a connection on the principal fibre bundle using horizontal subspaces as well as definition of a connection on the principal fibre bundle using lie algebra-valued differential forms.
