Implicit formula for the Levi-Civita connection

differential-geometry differential-forms riemannian-geometry connections lie-derivative

Solution 1:

Using the invariant formula for the exterior derivative, we have that

$$\textrm d \theta_X (Y, Z) = Y \cdot \theta_X (Z) - Z \cdot \theta_X (Y) - \theta_X ([Y, Z]) = Y \cdot g(X, Z) - Z \cdot g(X, Y) - g(X, [Y, Z]) .$$

Similarly, using the properties of the Lie derivative, we have

$$(L_X g) (Y, Z) = X \cdot g(Y, Z) - g([X, Y], Z) - g(Y, [X, Z]) .$$

Putting all these together we obtain

$$(L_X g) (Y, Z) + \textrm d \theta_X (Y, Z) = X \cdot g(Y, Z) + Y \cdot g(Z, X) - Z \cdot g(X, Y) - g([X, Y], Z) - g([Y, Z], X) - g([X, Z], Y) .$$

This last expression is known to be equal to $2 g (\nabla_Y X, Z)$ by Koszul's formula (warning: the link to Wikipedia points to formulae that obtain $\nabla_X Y$, not $\nabla_Y X$, so a sign will differ!).

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