- Thread starter
- Moderator
- #1

- Jan 26, 2012

- 995

-----

**Problem**: Show that an affine connection $\nabla$ is compatible to the Riemannian metric $\langle\cdot,\cdot\rangle$ if and only if, for any curve $c:I\rightarrow M$, and for any pair of vector fields $V$, $W$ along $c$, we have

$$\frac{\,d}{\,dt} \left\langle V,W\right\rangle= \left\langle\frac{\,DV}{\,dt},W\right\rangle + \left\langle V,\frac{\,DW}{\,dt}\right\rangle$$

where $\dfrac{D}{dt}$ denotes the covariant derivative.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!