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