Blaschke’s Problem for Timelike Surfaces

Abstract

A classification of timelike spherical congruences which induce plus or minus the same conformal structure on a timelike surface are classified. In the positive definite setting Hertrich-Jeromin [3] calls this Blaschke’s problem. This classification uses the Mobius geometry for timelike surfaces. The setting differs from the positive definite case because the projectivized light cone in \({\mathbb {R}}^5_1\) can be identified with the sphere S3, while the same construction in \({\mathbb {R}}^5_2\) yields a compact space homeomorphic to \(\frac{s^{2} \times s^{1}}{s^{0}}\), which is not homeomorphic to the Lorentzian sphere \(S^3_1\).