Lifts
There are various ways to lift objects on M into objects on TM. For example, if c is a curve in M, then c' (the tangent of c) is a curve in TM. Let us point out that without further assumptions on M (say, a Riemannian metric), there is no similar lift into the cotangent bundle.
The vertical lift of a function is the function defined by, where is the canonical projection.
Read more about this topic: Tangent Bundle
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