The Bundle of Penrose Limits: weighted geometry, contact structure, integrable systems, and Cartan-Ehresmann soldering

The classical Penrose limit associates to an affinely parameterized null geodesic a plane-wave metric obtained by anisotropic blow-up, but its usual coordinate presentation obscures the intrinsic geometry behind the construction. In this lecture, I will explain a bundle-theoretic reformulation of the Penrose limit in which the basic object is not an ordinary tangent-space limit, but a graded model attached to the null filtration along a null geodesic.  The familiar Penrose scaling appears as the Euler dilation of this graded geometry.

I will show how the space of unparameterized null geodesics carries the contact structure needed to interpret the weight-two Penrose direction, and how the 1-jets of contact scale organize the resulting plane-wave germs into an intrinsic Penrose bundle, with a polarized refinement after the choice of a Lagrangian subspace. Restricting this bundle to a Legendrian family gives canonical polarized data and leads to a local Rosenization. I will then describe a Cartan-Ehresmann viewpoint on this geometry, modeled on the Euler-Arnold flow in the symplectic loop group: first in its partial form governed by the Jacobi/matrix Korteweg-de Vries system, and then in a fully soldered form in which the derivative of the Legendrian polarization supplies the missing multidirectional input. The result is a geometric framework in which weighted Penrose limits, Jacobi fields, and integrable deformations of plane waves fit into a single picture.