On the existence and prolongation of infinitesimal isometries on special sub-Riemannian manifolds

TL;DR

This paper investigates conditions under which local infinitesimal isometries can be constructed and extended on special sub-Riemannian manifolds, focusing on the role of $ ext{is}^*$-regular and $ ext{is}$-regular points.

Contribution

It introduces modified notions of regular points for special sub-Riemannian manifolds and establishes criteria for constructing and prolonging infinitesimal isometries.

Findings

Conditions for constructing local infinitesimal isometries

Criteria for prolonging isometries to larger domains

Extension of Nomizu's notions to sub-Riemannian context

Abstract

In the present paper we deal with (local) infinitesimal isometries of special sub-Riemannian manifolds (a contact and oriented sub-Riemannian manifold is called special if the Reeb vector field is an isometry). The objective of the paper is to find some conditions on such manifolds which allow one to construct, locally around a given point, infinitesimal isometries and then, possibly, to prolong them onto bigger domains. The mentioned conditions are related to the so-called $\is^*$-regular and $\is$-regular points, the notions introduced by Nomizu in [22] in the Riemannian setting and slightly modified by the author.