General Relativity: An Introduction for Physicists · Problem 3.4

Vector calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · affine-connection-basis-vectors

Problem

Show that the affine connection can be written as

\Gamma^b{}_{ac} = e^b \cdot \partial_c e_a

. Show further that, in a torsionless manifold,

\partial_c e_a = \partial_a e_c