Thread: Question about linear connection and covariant derivative along curves

Hello,

Let M be a Riemannian Manifold, Levi-Civita-Connection and the induced covariant derivative along curves.
Let be a curve and an intervall, Z a vector field along c.

Can someone help me find out why the following statement is true?


(The left is the covariant derivative along c and the right is the covariant derivative along ).

I already know that if X and Y are vector fields along c and if then .

Regards,
engmaths

Suppose L is open so that for any t0 in L, there is a small r such that (t0-r, t0+r) is in L.
If L is not open, for example L=[a,b), we can only define D/dt(Z|L) from the right side of a, thus there is a small r such that [a, a+r) is in L so the statement still holds.

Hello xxp9,

I'm not quite sure if you understood my problem.

If we assume that L is open, why is the statement true then?

Somehow I don't see a way to compare the left (the covariant derivative along c) and the right (the covariant derivative along ).

As you said if two vector fields are identical in an interval around a point along the curve,, they covariant derivative are identical at that point.

I got another question concerning this topic:

Assume you have a smooth curve c:J->M that admits an extension to a smooth curve definied on a larger intervall.
Now, is then every vector field along c extendible to a vector field along the extension of c?

since we can always express a vector field in terms of its components, which are smooth functions, based on the coordinates vectors, the question is equivalent to:
Given a smooth function defined on [a, b], can we extend it to a larger interval?
The answer is yes.