- #1

- 8,942

- 2,933

So let's consider a diffeomorphism generated by a vector field ##V##. If ##X## is a point on our manifold, then we can define a parametrized path ##Y_X(\lambda)## via:

##Y_X(0) = X##

##\frac{d}{d\lambda} Y_X(\lambda) = V|_{Y_X(\lambda)}##

So far, ##X## and ##Y## are points on the manifold, not coordinates, but now I'd like to switch to a coordinate system. I'll just use ##X^\alpha## and ##Y^\alpha## to mean the coordinates of points ##X## and ##Y##, and use ##V^\alpha## to mean the components of ##V## in the coordinate basis for our coordinate system.

The first uncertainty on my part is whether the following is true:

If we want to know the coordinates for the point ##Y_X(\lambda)##, I think it's just a matter of integration:

##Y^\alpha_X(\lambda) = Y^\alpha_X(0) + \int V^\alpha|_{Y_X(\lambda)} d\lambda##

## = X^\alpha + \int V^\alpha|_{Y_X(\lambda)} d\lambda##

Assuming that is correct, what I want to compute next is the dependence of ##Y^\alpha_X(\lambda)## on ##X##. At this point, my first inclination would be:

##\frac{\partial Y^\alpha_X(\lambda)}{\partial X^\beta} = \frac{\partial X^\alpha}{\partial X^\beta}+ \int \frac{\partial V^\alpha}{\partial Y^\mu}|_{Y_X(\lambda)} \frac{\partial Y^\mu_X(\lambda)}{\partial X^\beta} d\lambda##

My uncertainty is with the expression ##\frac{\partial V^\alpha}{\partial Y^\mu}##. Should that be ##\nabla_\mu V^\alpha## rather than the partial derivative? My feeling is that it shouldn't be, since the concept of diffeomorphism is independent of whether there is a connection, or not.