 Vertical vector field of a surface unit tangent bundle

ID: vertical-vector-field-of-a-surface-unit-tangent-bundle

On an oriented Riemannian surface, the vertical vector field generates positive rotation in each fibre of its unit tangent bundle:

V f (x, v) = \partial_{t} f (x, cos t v + sin t i v) ∣_{t = 0}

. Hence

V (v) = i v

and

V (i v) = - v

. In the canonical frame,

[V, X] = H

[V, H] = - X

and

[X, H] = K V

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