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
Famous quotes containing the words canonical, field and/or bundle:
“If God bestowed immortality on every man then when he made him, and he made many to whom he never purposed to give his saving grace, what did his Lordship think that God gave any man immortality with purpose only to make him capable of immortal torments? It is a hard saying, and I think cannot piously be believed. I am sure it can never be proved by the canonical Scripture.”
—Thomas Hobbes (1579–1688)
“There is a call to life a little sterner,
And braver for the earner, learner, yearner.
Less criticism of the field and court
And more preoccupation with the sport.”
—Robert Frost (1874–1963)
“We styled ourselves the Knights of the Umbrella and the Bundle; for, wherever we went ... the umbrella and the bundle went with us; for we wished to be ready to digress at any moment. We made it our home nowhere in particular, but everywhere where our umbrella and bundle were.”
—Henry David Thoreau (1817–1862)