Conformal Lorentzian metrics on the spaces of curves and 2-surfaces

We show how, beginning with the space of curves in R2 and the space of 2-surfaces in R3, one can define conformal Lorentzian three- and four-dimensional metrics on certain special subspaces. It is conjectured that for the case of curves in R2 this is how Wunschmann obtained his well-known equation. Generalizing the argument to the case of 2-surfaces, leads to a generalized Wunschmann equation. We point out that all conformal Lorentzian three- and four-dimensional metrics (thus including Einstein metrics) can be obtained in this manner.