The angle between two vectors remain the same after the parallel translation and Riemannian connection.

Question




Let $M$ be a Riemannian manifold, $\nabla$ be a symmetric affine connection, prove:
If the angle between two vectors remain the same after the parallel translation along any curve, then $\nabla$ is a Riemannian connection.


 I have tried choosing an orthonormal basis of $p\in M$, but parallel translation doesn't keep the scale of the vectors, thus the scales of basis can't remain the same, as a result we can't use the classical method described in Do Carmo's book. So can someone provide a solution to this problem?

Thanks in advance.