Analogues of complex geometry

Analogues of Complex Geometry

We prove that there is no hope of trying to invent pseudoholomorphic theories of anything other than curves, even if one allows more general objects than almost complex manifolds. The hard part is dispensing with theories of pseudoholomorphic hypersurfaces; hypersurface theories exist “microlocally” (in abundance) but not “locally.”

The complex geometry of

Lagrange Geometry via Complex Lagrange Geometry

Asking that the metric of a complex Finsler space should be strong convex, Abate and Patrizio ([1]) associate to the real tangent bundle a real Finsler metric for which they analyze the relation between Cartan (real) connection of the obtained space and the real image of Chern-Finsler complex connection. Following the same ideas, in the present paper we shall deal with the more general case of ...

Lipschitz Geometry of Complex Curves

We describe the Lipschitz geometry of complex curves. For the most part this is well known material, but we give a stronger version even of known results. In particular, we give a quick proof, without any analytic restrictions, that the outer Lipschitz geometry of a germ of a complex plane curve determines and is determined by its embedded topology. This was first proved by Pham and Teissier, b...

Coolidge on Complex Geometry