Show that a parallel field has constant length.

Question

Show that a parallel field has constant length (Riemannian-geometry). It is true for all connections?

Answer 1

The Levi-Civita connection preserves the metric. That is, $\nabla g = 0$.

What does $\nabla g$ mean? It's defined such that $\nabla (g(X,Y)) = (\nabla g) (X,Y) + g(\nabla X, Y) + g(X, \nabla Y)$.

So the statement that $\nabla g = 0$ is the statement that, for any vector field $X$, we have $X g(Y,Z) = g(\nabla_X Y, Z) + g(Y, \nabla_X Z)$. Sometimes this is taken as an axiom for the Levi-Civita connection instead of $\nabla g = 0$, but they're equivalent.

In any case, to the problem at hand. Let $Y$ be parallel, i.e. $\nabla Y = 0$, i.e. $\nabla_X Y = 0$ for all $X$.

Then $\nabla (g(Y,Y)) = g(\nabla Y,Y) + g(Y,\nabla Y) = 0 + 0 = 0$. Thus $g(Y,Y)$ is constant.

If you're dealing with a different sort of field, the same sort of derivation holds.

This is not true for general connections; they must be metric, i.e. $\nabla g = 0$.