Riemannian metric on SL(2,R)

Every Lie group is like a Universe. Now it is time for us to play with the cosmology of SL(2,R). In Real magic – space-time in Lie algebra we have already started this game – we have defined a natural “metric” on the Lie algebra sl(2,R). We have defined scalar product of any two vectors tangent to the group at the group identity. And we have seen that this scalar product is similar to that of Minkowski space-time, except that with only two space (and one time) dimensions.

Riemannian metric, on the other hand, is defined when we have scalar product of tangent vectors at every point. In recent posts we were playing with Riemannian metric on the upper half-plane, which is a homogeneous space for the group, of only two dimensions – the cross section of the torus. Now we want to define Riemannian metric on the whole torus – an interesting extension.

We already have the metric at one point – at the group identity. Can we extend this definition to the whole torus, and do it in a natural way?

Of course we can. Because we are on the group. Here it is how it is being done. Suppose we have two vectors tangent at some group point g. I am using the letter g now, but g here is another notation for a real matrix A of determinant one, an element of SL(2,R). That is we have two paths \gamma_1(t),\gamma_2(t) with \gamma_1(0)=\gamma_2(0)=g. Our two tangent vectors, say \xi_1,\xi_2, are vectors tangent to \gamma_1(t) and \gamma_2(t) at g, they are represented by matrices

(1)   \begin{equation*}\xi_i=\frac{d\gamma_i(t)}{dt}|_{t=0}, \quad (i=1,2).\end{equation*}

The matrices \xi_1,\xi_2 are not in the Lie algebra. For instance, in our case, they will not be (in general) of trace zero. That is because at t=0 the two paths are not at the identity. But we can shift them to the identity. The paths g^{-1}\gamma_i(t) at t=0 are at the identity. Thus even if \xi_i are not in the Lie algebra, g^{-1}\xi_i are. We can use this fact and define the scalar product at g as follows:

(2)   \begin{equation*}(\xi_1,\xi_2)_g=(g^{-1}\xi_1,g^{-1}\xi_2)_e.\end{equation*}

I am using the symbol e rather than the unit matrix I to denote the group identity here, because the construction above is quite general, is being used for any Lie group, not just for SL(2,R).

Thus once we have scalar product defined in the Lie algebra, we have it defined everywhere.

Of course one can ask: why not use another definition, with right shifts? What is wrong with

(3)   \begin{equation*}(\xi_1,\xi_2)_g=(\xi_1g^{-1},\xi_2g^{-1})_e.\end{equation*}

Nothing is wrong. The scalar product at the identity that we have defined in Real magic – space-time in Lie algebra has the property of invariance expressed there in Eq. (11). It is a homework to show that using this property we can prove that the two definitions above lead to the same Riemannian metric!

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