Inverting the parallel transport via path-ordered exponential

In Riemann geometry one can formally solve the parallel transport equation

$$ dot{v}^mu + Gamma^mu_{rhosigma} , u^rho , v^sigma = 0 $$

of a vector $v$ along a curve with unit tangent vector $u^mu = dot{x}^mu$ using the path-ordered exponential

$$ P^mu_nu(s,0) = left( text{P exp} – int_0^s ds , Gamma , u right)^mu_nu $$

$$ v^mu(s) = P^mu_nu(s,0) , v^nu(0) $$

Suppose we have

$$ langle v(s), w(s) rangle = langle P(s,0) , v(0), P(s,0) , w(0) rangle $$

with

$$ langle v, w rangle = g_{munu} , v^mu , w^nu $$

Question: can one show that the path-ordered exponentials cancel?

$$ langle v(s), w(s) rangle = langle v(0), w(0) rangle $$