# The reflection map and infinitesimal deformations of sphere mappings

## TL;DR

This paper uses the reflection map to simplify nondegeneracy conditions for sphere maps and investigates the structure and bounds of their infinitesimal deformations, providing new characterizations.

## Contribution

It introduces a new application of the reflection map to analyze infinitesimal deformations and bounds their dimension for nondegenerate sphere maps.

## Key findings

- Bound on the dimension of infinitesimal deformations
- Characterization of the homogeneous sphere map
- Simplified nondegeneracy conditions

## Abstract

The reflection map introduced by D'Angelo is applied to deduce simpler descriptions of nondegeneracy conditions for sphere maps and to the study of infinitesimal deformations of sphere maps. It is shown that the dimension of the space of infinitesimal deformations of a nondegenerate sphere map is bounded from above by the explicitly computed dimension of the space of infinitesimal deformations of the homogeneous sphere map. Moreover a characterization of the homogeneous sphere map in terms of infinitesimal deformations is provided.