Re: gradient field/geodesics

In news:<1146005662.390315.139010@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>
schrieb <djunkmayl1@xxxxxxxxx>:
Is there anyone who could give me please help on the question:

Let M be a Riemannian manifold and let f be a smooth function on M
whose gradient vector field G=grad(f) has |G|=1 everywhere. Show that
the integral curves of G are geodesics.

For every vector field X on M, there holds:

<D_G G, X> = G (<G,X>) - <G, D_G X>
= G X (f) - <G, D_G X>
= X G (f) + [G,X](f) - <G, D_G X>
= X(<G,G>) + <G, [G,X] - D_G X>
= X(<G,G>) + <G, D_X G>
= 1/2 X(<G,G>)
= 1/2 X(1) = 0.

.