Curvature through the Looking-Glass: Reflections, Cross Ratios, and the Geometry of Plane Waves

The classical cross ratio of four points on a line is a small miracle: it is built from the simplest projective data, it survives every Möbius transformation, and when you let the points move it leads straight to the Schwarzian derivative---an invariant that detects when a motion is ``projectively trivial.''

This talk is about the same story, but with the points replaced by subspaces.  Given four suitably transverse subspaces of a vector space, one can form an operator-valued cross ratio by composing the obvious projections; in coordinates it reduces to a familiar fractional-linear expression, and its symmetries are controlled by a handful of reflection identities (the higher-dimensional analogue of the anharmonic group).  The first part of the talk is concrete and largely linear-algebraic---you can do most of it with block matrices and a good picture of graphs of linear maps.

Next we let the subspaces move.  By comparing cross ratios at nearby times and taking a differential limit, we obtain a Schwarzian-type invariant for curves in a Grassmannian: it measures exactly how far the curve is from being generated by a fractional-linear change of viewpoint.  Finally, we explain why plane gravitational waves are a natural home for this construction: the Jacobi equation along a null geodesic produces a canonical curve of Lagrangian subspaces, and in that setting the generalized Schwarzian becomes a clean signature of the wave, packaging its tidal curvature profile.  Proofs will be short and computational; the curvature identification is presented as the closing dictionary.