In the "imaginary geometry I:interacting sles" we work with Imaginary geometry:
"In the language of differential geometry, an imaginary geometry is a two dimensional manifold endowed with a torsion-free affine connection whose holonomy group consists entirely of dilations (c.f. ordinary Riemannian surfaces, whose Levi-Civita holonomy groups consist entirely of rotations), and straight lines are geodesic flows of the connection.
The connection endows the manifold with a conformal structure, and by the uniformization theorem one can conformally map the geometry to a planar domain on which the geodesics are determined by some function h."
Any references from differential geometry will be greatly appreciated.