- #1

Pentaquark5

- 17

- 2

Let ##\gamma(s)## be a smooth curve, and ##l_0 \in T_p\mathcal{M}## the tangent vector at ##\gamma(s_0)=p##. Then we can parallel transport ##l_0## along ##\gamma## by using

$$ \nabla_{\gamma'(s)}l(s)=0 .$$

However, I'm having trouble applying this knowledge to the following case (quote from my script):

Let ##e_0:=\dot{\gamma}## and ##e_1 \perp e_0## be unit vectors. Let further ##S## be the ##2##-dimensional submanifold of ##\mathcal{M}## obtained by shooting null-geodesics with initial direction ##l(0):=e_0+e_1## from all points along the curve ##\gamma##.

Imposing affine parameterisation, this defines on ##S## a vector field ##l## tangent to those geodesics, solution of the equation

$$\nabla_l l=0.$$

I interpret this equation as the parallel transport of ##l## along ##l##, but that does not really coincide with the description given in my script.

Could anybody help me make sense of this equation? Thanks