The 4-velocity in contravariant form is given by $$V^\mu=\frac{dx^\mu}{d\tau}$$ for some general co-ordinates $x^\mu$ and proper time $\tau$.

Is the 4-velocity in covariant form given by

$$V_\nu=V^\mu g_{\mu\nu}=\frac{dx^\mu}{d\tau}g_{\mu\nu}=\frac{dx_\nu}{d\tau}?$$