Sub-twistor metrics

TL;DR

This paper introduces and characterizes sub-twistor metrics on hyperk"ahler manifolds, showing they are Finsler and providing explicit descriptions of their norms within the framework of metric geometry.

Contribution

It defines sub-twistor metrics, proves they are sub-Finsler, and extends the understanding of distance functions on hyperk"ahler manifolds in metric geometry.

Findings

Sub-twistor metrics are Finsler.

Explicit descriptions of Finsler norms are provided.

Sub-conic metrics are shown to be sub-Finsler.

Abstract

We consider a natural distance function on the period space of a hyperk\"ahler manifold associated to non-holonomic constraints imposed by twistor lines. These metrics were introduced by Verbitsky in the context of the global Torelli theorem for hyperk\"aler manifolds. We show that they are Finsler and explicitly describe their Finsler norms. To achieve this goal we consider a class of distance functions (called here sub-conic metrics) that slightly generalize sub-Riemann metrics. The main technical result is the statement that every sub-conic metric is sub-Finsler. The content and methods of the paper lie within basic metric geometry. The complex geometrical background and motivation are isolated in the appendix.