Canonical Vector Field On Tangent Bundle
On every tangent bundle TM one can define a canonical vector field . If (x, v) are local coordinates for TM, the vector field has the expression
Alternatively, consider to be the scalar multiplication function . The derivative of this function with respect to the variable at time is a function, which is an alternative description of the canonical vector field.
The existence of such a vector field on TM can be compared with the existence of a canonical 1-form on the cotangent bundle. Sometimes V is also called the Liouville vector field, or radial vector field. Using V one can characterize the tangent bundle. Essentially, V can be characterized using 4 axioms, and if a manifold has a vector field satisfying these axioms, then the manifold is a tangent bundle and the vector field is the canonical vector field on it. See for example, De León et al.
Read more about this topic: Tangent Bundle
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