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Section 3.8 Lie Group Connection

The Killing form introduced in Section 3.6 acts as an inner product on the Lie algebra, that is, on the tangent space to the Lie group at the identity. As discussed in Section 3.2, vectors in the Lie algebra extend to left-invariant vector fields on the Lie group. Thus, the Killing form extends to a metric on the Lie group, which is constant on left-invariant vector fields by construction. Given a metric, we can construct its Levi–Civita connection and curvature.

The Levi–Civita connection is uniquely determined by the requirements that it be (a connection that is) metric compatible and torsion free. The connection \(\nabla\) is metric compatibile if

\begin{equation} Z\bigl(B(X,Y)\bigr) = \nabla_Z\bigl(B(X,Y)\bigr) = B\bigl(\nabla_ZX,Y\bigr)+B\bigl(X,\nabla_ZY)\tag{3.8.1} \end{equation}

that is, if the derivative of “dot” is zero (\(\nabla_ZB=0\)). However, we know from (3.6.14) that the Killing form is constant when acting on left-invariant vector fields, so that both sides of (3.8.1) are in fact zero. The connection is torsion free if

\begin{equation} [\nabla_X,\nabla_Y]f = \nabla_{[X,Y]}f = [X,Y](f)\tag{3.8.2} \end{equation}

for any function \(f\text{.}\) In both expressions, we have used the fact that

\begin{equation} \nabla_X f = X(f)\tag{3.8.3} \end{equation}

for any connection, that is, the covariant derivative of a function is just its directional derivative (which is \(\vf{X}\cdot\grad f\) in the language of vector calculus).

Putting these properties together, we can compute  1 

\begin{align} B(\nabla_XY,Z) \amp= B([X,Y],Z) + B(\nabla_YX,Z)\notag\\ \amp= B([X,Y],Z - B(X,\nabla_YZ)\notag\\ \amp= B([X,Y],Z - B(X,[Y,Z]) - B(X,\nabla_ZY)\notag\\ \amp= B([X,Y],Z - B(X,[Y,Z]) + B(\nabla_ZX,Y)\notag\\ \amp= B([X,Y],Z - B(X,[Y,Z]) + B([Z,X],Y) + B(\nabla_XZ,Y)\notag\\ \amp= B([X,Y],Z - B(X,[Y,Z]) + B([Z,X],Y) - B(Z,\nabla_XY)\tag{3.8.4} \end{align}

which we can rearrange to

\begin{equation} 2B(\nabla_XY,Z) = B([X,Y],Z) - B([Y,Z],X) + B([Z,X],Y) = B([X,Y],Z)\tag{3.8.5} \end{equation}

where we have used the cyclic property (3.6.6) of the Killing form to cancel the last two expressions. Finally, since \(Z\) is arbitrary (and \(B\) is nondegenerate), we can conclude that

\begin{equation} \nabla_XY = \frac12 [X,Y] .\tag{3.8.6} \end{equation}
In general, this derivation yields the Koszul formula for the Levi–Civita connection.