Complete Lift for Metric and Connection on Banach Manifolds

Complete Lift for Metric and Connection on Banach Manifolds

In this paper for a given Banach manifold M firstly we define the complete lift for metric g on M (or 2-forms), then we lift connection ∇ on M to ∇c on TM. The main goal for this lifting is showing that if connection ∇ and metric g are compatible then their complete lift are too. Also the geodesic for ∇c is the complete lift of geodesic for ∇. Afterward we focus on T1M with a generalized Fr´echet manifold structure and we will prove that any connection ∇ on M can be lifted to a connection ∇c1 on T∞M. Finally we show that there is the unique geodesic for conjoin any arbitrary two point on T∞M.