Connections in GL(2,R)

Question

I want to prove that there exists a unique affine connection on $GL(2,\mathbb{R})$ such that all left-invariant vector fields are parallel, and find its torsion.

Answer 1

Given any framing of a manifold $M^n$, i.e. a set of vector fields $X_1, \ldots X_n$ which are a basis of the tangent space of $M$ at each point, and an arbitrary set of $n^3$ smooth functions $\Gamma^k_{ij}$ on $M$, a connection is defined uniquely by setting $\nabla_{X_i}X_j=\sum_k \Gamma^k_{ij} X_k$, then extending to arbitrary vector fields by linearity and the Leibniz rule.

For a Lie group, a basis of the space of left-invariant vector fields is such a framing, hence if one requires them to be parallel, i.e. $\nabla X_i=0$, this defines a connection on the group uniquely, by the above remark.

(Note that, by the Leibniz rule, if a particular basis of left-invariant vector fields is parallel, then any basis of such vector fields is parallel, since the change of basis is given by constant functions, hence the above definition is independent of the particular basis of left-invariant vector fields chosen).

As for the torsion tensor, it is given by the formula $T(X,Y)=\nabla_XY-\nabla_Y(X)-[X,Y],$ hence for left-invariant vector fields $T(X,Y)=[X,Y].$ It follows that the connection is torsionless if and only if the group is commutative.

In your case the group is not commutative, hence the connection has torsion. You can pick a nice basis of the Lie algebra ($2\times 2$ matrices), calculate the commutation relations (they are very simple) and write down explicitly the torsion in terms of them.

Answer 2

Well I am sorry that some (not all) of the comments you received are insensitive. My Differential Geometry is quite rusty but my idea is to define a covariant differentiation, $\frac{dV}{ds}$. If $X_1, ... X_n$ is a basis of the left invariant vector fields then any vector field is of the form $$X=f_1X_1+\cdots +f_nX_n$$ So $$\frac{dX}{ds}=\frac{df_1}{ds}X_1+\cdots \frac{df_n}{ds}X_n$$ I believe that the covariant differentiation will define a connection. It should be clear that this is the only possible covariant derivative with the property you specify.