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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.

Resources:

1)http://math.mit.edu/~sheffield/geodesics.html

2)https://arxiv.org/abs/1201.1496

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