Geodesics on locally homogeneous affine surfaces

Peter B. Gilkey. University of Oregon.

Resumo:

We examine questions of geodesic completeness in the context of locally homogeneous affine surfaces. Any locally homogeneous affine surface has a local model $\mathcal{M}$ where either the Christoffel symbols take the form $\Gamma_{ij}{}^k$ are constant and the underlying space is $\mathbb{R}^2$ (Type-A) or the Christoffel symbols take the form $\Gamma_{ij}{}^k=C_{ij}{}^k/x^1$ where the underlying space is $\mathbb{R}^+\times\mathbb{R}$ (Type-B).

The model space $\mathcal{M}$ is said to be ESSENTIALLY GEODESICALLY COMPLETE if there does not exist a complete locally homogeneous surface modeled on $\mathcal{M}$ which is geodesically complete. Up to linear equivalence, there are exactly 3 models which are geodesically incomplete but not essentially geodesically incomplete. We classify all the geodesically complete models of Type-A and present some partial results concerning Type-B models.

This is joint work in progress with Gabriela Beneventano, Daniela Dascanio, Pablo Pisani, and Eve Mariel Santangelo (Universidad Nacional de La Plata, Argentina).

© Peter B. Gilkey.