Geodesics: Constant Velocity & Affine Parameterization

[latentcorpse]

Affinely parameterised geodesics satisfy $\nabla_XX=0$
Why does this mean they have constant velocity?

Thanks.

 

[Tangent87]

If we parametrize the geodesic by $$\lambda$$ then the velocity is given by $$\frac{dX^a(\lambda)}{d\lambda}=\dot{X}^a$$

Thus, $$\nabla_{\lambda}\dot{X}^a=\dot{X}^b\nabla_{b}\dot{X}^a=\dot{X}^b\left(\partial_b \dot{X}^a+\Gamma_{bc}^a \dot{X}^c\right)=\ddot{X}^a+\Gamma_{bc}^a\dot{X}^b\dot{X}^c=0$$ since it is the geodesic equation. Therefore the velocity is constant along the geodesic.