Conformally covariant differential symmetry breaking operators for a vector bundle of rank 3 over S^3

TL;DR

This paper classifies all conformally covariant differential symmetry breaking operators between vector bundles over S^3 and line bundles over S^2, providing explicit conditions for their existence.

Contribution

It offers a complete classification and explicit construction of symmetry breaking operators for vector bundles over spheres, with precise parameter conditions.

Findings

Explicit classification of symmetry breaking operators

Necessary and sufficient conditions for operator existence

Construction of operators for specific parameter tuples

Abstract

We construct and give a complete classification of all the differential symmetry breaking operators D_{{\lambda},{\nu}}^m : C^\infty(S^3, V^3_{\lambda}) \rightarrow C^\infty(S^2,L_{m,{\nu}}), between the spaces of smooth sections of a vector bundle of rank 3 over the 3-sphere V^3_{\lambda} \rightarrow S^3, and a line bundle over the 2-sphere L_{m,{\nu}} \rightarrow S^2. In particular, we give necessary and sufficient conditions on the tuple of parameters ({\lambda}, {\nu}, m) for which these operators exist.