Inner product of parallel transported vectors along a curve

[WannabeNewton]

Homework Statement

Given that the vectors $$\underset{A}{\rightarrow}$$ and $$\underset{B}{\rightarrow}$$ are parallel transported along a curve:
$$\triangledown _{\underset{l}{\rightarrow}}A = 0$$
$$\triangledown _{\underset{l}{\rightarrow}}B = 0$$

Show that g(A,B) = constant along the curve

Homework Equations

The Attempt at a Solution

I just have one major question for this problem: can one say that if the covariant derivative of g(A,B) = 0 then g(A,B) = constant i.e. if
$$\triangledown _{\underset{l}{\rightarrow}}(g(A,B)) = \triangledown _{\underset{l}{\rightarrow}}(g_{\alpha \beta }A^{\alpha }B^{\beta }) = A^{\alpha }B^{\beta }\triangledown _{\underset{l}{\rightarrow}}g_{\alpha \beta } + g_{\alpha \beta }B^{\beta }\triangledown _{\underset{l}{\rightarrow}}A^{\alpha} + g_{\alpha \beta }A^{\alpha }\triangledown _{\underset{l}{\rightarrow}}B^{\beta } = 0$$

then g(A,B) = const. for the curve?

 

[Dick]

Sure, that's right. Now what's the covariant derivative of g?