Can Lie Derivatives be Defined Without Dragging on Manifolds?

[wdlang]

suppose there is a vector field V on a manifold M

V generates a flow on M

suppose \gamma(t) is an integral curce in this flow

now there is another vector field W on M

why not define the lie derivative of W with respect to V as the limit of the divide

(W(\gamma+\delta \gamma)-W(\gamma))/\delta \gamma

here the difference is taken by components

i think this is very natural from our experience in the calculus course in undergraduate.

why we need to drag?

 

[quasar987]

The difference W(\gamma+\delta \gamma)-W(\gamma)) does not make sense since W(\gamma+\delta \gamma) and W(\gamma)) are elements of different vector spaces.

 

[Fredrik]

The expression $\gamma+\delta\gamma$ (which I assume means $\gamma(t)+\delta\gamma(t)$) doesn't make sense either, since a manifold doesn't necessarily have a vector space structure. For most manifolds, both the + and the multiplication in $\delta\gamma(t)$ are undefined.