Question about linear connection and covariant derivative along curves

Hello,

Let M be a Riemannian Manifold, $\displaystyle \nabla$ Levi-Civita-Connection and $\displaystyle \frac{D}{dt}$ the induced covariant derivative along curves.

Let $\displaystyle c:J\rightarrow M$ be a curve and $\displaystyle L\subset J$ an intervall, Z a vector field along c.

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

$\displaystyle (\frac{D}{dt}Z)|_L=\frac{D}{dt}(Z|_L)$ (The left $\displaystyle \frac{D}{dt}$ is the covariant derivative along c and the right $\displaystyle \frac{D}{dt}$ is the covariant derivative along $\displaystyle c|_L$).

I already know that if X and Y are vector fields along c and if $\displaystyle X|_{(t_0-\varepsilon,t_0+\varepsilon)}=Y|_{(t_0-\varepsilon,t_0+\varepsilon)}$ then $\displaystyle (\frac{D}{dt}X)(t_0)=(\frac{D}{dt}Y)(t_0)$.

Regards,

engmaths

Re: Question about linear connection and covariant derivative along curves

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.

Re: Question about linear connection and covariant derivative along curves

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 $\displaystyle \frac{D}{dt}$ (the covariant derivative **along c**) and the right $\displaystyle \frac{D}{dt}$ (the covariant derivative **along $\displaystyle c|_L$**).

Re: Question about linear connection and covariant derivative along curves

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.

Re: Question about linear connection and covariant derivative along curves

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?

Re: Question about linear connection and covariant derivative along curves

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.