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?


this is in between. The second is the definition of the bracket, and requires very little.

The first is the definition of a torsion free connection. This also does not require a metric, certainly not the positive definite kind. In physics, they often use connections that have torsion, meaning the equation is false for some useful connections.

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.

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    $\begingroup$ Oh I see - the Levi-Civita connection is defined so that this identity holds. $\endgroup$ Jul 18 '15 at 21:42

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