Geometric features of Vessiot-Guldberg Lie algebras of conformal and Killing vector fields on \(\mathbb{R}^2\)

Overview

Translating or rotating a figure in the Euclidean plane leaves its lengths and angles unchanged. Uniformly enlarging it changes the lengths but preserves the angles. These familiar transformations illustrate a broader idea: we can understand a family of transformations through the geometric structure it preserves.

Infinitesimally, a transformation is described by a vector field: a direction of motion at each point. Following these directions produces its flow. A finite collection of vector fields \(X_1,\ldots,X_r\) spans a Lie algebra when their Lie brackets remain constant linear combinations of the same fields:

The bracket records how the flows interact when their order is exchanged. If a differential equation has a time-dependent vector field \(X(t)=\sum_i b_i(t)X_i\) within such an algebra, it is a Lie system, and the algebra is called a Vessiot-Guldberg Lie algebra. This finite algebraic structure gives us a way to study the equation's dynamics geometrically.

A metric \(g\) encodes the local geometry. Its change along a flow is measured by the Lie derivative \(\mathcal{L}_X g\). A Killing vector field leaves the metric unchanged; a conformal vector field allows a local rescaling:

For the Euclidean metric \(g=dx\otimes dx+dy\otimes dy\), rotation is generated by \(R=-y\,\partial_x+x\,\partial_y\), with \(\mathcal{L}_R g=0\). Dilation is generated by \(D=x\,\partial_x+y\,\partial_y\), with \(\mathcal{L}_D g=2g\). Rotation preserves lengths, while dilation rescales them. The same distinction can be expressed for general pseudo-Riemannian metrics.

In this paper, we locally classify the finite-dimensional Lie algebras of conformal and Killing vector fields on \(\mathbb{R}^2\) for arbitrary pseudo-Riemannian metrics. In two dimensions, every such metric is locally conformally flat, which lets us relate the classification to that of vector-field Lie algebras on the plane. We then study the geometric structures associated with these algebras.

One useful bridge from algebra to geometry is a quadratic Casimir: an algebraic expression invariant under the algebra's action. Replacing its generators with vector fields gives an invariant tensor. When this tensor is nondegenerate, its inverse defines a metric for which the vector fields are Killing. We also examine invariant distributions, which describe preserved directions, and induced symplectic structures that support a Hamiltonian description of the dynamics.

We apply these constructions to the Milne-Pinney equation and the projective Schrödinger equation on the Riemann sphere. The work grew out of my BSc thesis at the University of Warsaw. Its emphasis on preserved geometric structure also runs through my later space-folding work, where convexity becomes the central invariant.

@article{lewandowski2017geometric,
  title = {Geometric features of {Vessiot--Guldberg} Lie algebras of conformal and Killing vector fields on {$\mathbb{R}^2$}},
  author = {Lewandowski, M. M. and de Lucas, J.},
  journal = {Banach Center Publications},
  volume = {113},
  pages = {243--262},
  year = {2017},
  doi = {10.4064/bc113-0-13},
  url = {https://doi.org/10.4064/bc113-0-13},
  eprint = {1708.09375},
  archivePrefix = {arXiv},
  primaryClass = {math-ph}
}