# Levi-Civita connection on a vector space

let $$V$$ be vector space, and let $$\nabla$$ be the Levi-Civita connection wrt. a constant metric on $$V$$ (in the sense that the metric is the same at every point of $$V$$).

Let $$w$$ be a vector in $$V$$, and consider it a constant vector field on V, along some curve $$\gamma$$ on V.

Does it hold that

$$\nabla_{\dot{\gamma}_t}w = 0$$

?

• Yes, it does hold. – Berci Sep 27 at 10:57

Yes. This is the easiest possible case of a Riemannian manifold.

The Christoffel symbols $$\Gamma^i_{jk}$$ are 0 because all the derivatives of the metric are 0. So, in coordinates/components, $$\nabla_j w^i = \dfrac{\partial w^i}{\partial x^j} + \Gamma^j_{ik}w^k = 0$$ for every $$i,j$$ (and hence also for the vector $$\dot\gamma$$).