General Relativity: An Introduction for Physicists · Problem 3.11

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · vanishing-covariant-derivative-condition

Problem

Show that for a vector field with contravariant components

v^b

to have a vanishing covariant derivative

\nabla_a v^b

everywhere in a manifold, it must satisfy the relation

(\partial_b \Gamma^d{}_{ac} - \partial_c \Gamma^d{}_{ab} + \Gamma^e{}_{ac}\Gamma^d{}_{eb} - \Gamma^e{}_{ab}\Gamma^d{}_{ec})v^a = 0.

Hint: Use the fact that partial derivatives commute.