# Lie bracket is part of the intrinsic "geometry"? But I have seen it defined without a metric...?

I have seen two definitions of the Lie Bracket for a Riemannian manifold $(M,g)$.

One is this : $[X,Y] = D_X Y - D_Y X$, where $D$ stands for covariant differentiation. When written out, this seems to involve the Christoffel symbols. (This is in Thorpe's Elementary Topics in Differential Geometry.)

The other is this: $[X,Y] = XY - YX$ with $(XY - YX)(f) = (X(Y(f)) - Y(X(f))$. (And the thing about differentiation of Y along the flow of X.) (For instance, in Warner's Differential Manifolds.)

So my confusion is this: one expression seems to involve the metric, the other does not. How can I reconcile this?

The Riemannian metric comes in in the defining conditions for the Levi-Civita connection. with torsion free, we add on $$X \langle Y,Z \rangle = \langle \nabla_X Y,Z \rangle + \langle Y, \nabla_X Z \rangle$$ which greatly resembles the product rule in beginning calculus.