Holomorphic isometric embeddings of the projective line into quadrics

TL;DR

This paper classifies holomorphic isometric embeddings of the projective line into quadrics, describing their moduli spaces and proving the rigidity of the real standard map.

Contribution

It generalizes the do Carmo--Wallach theorem to describe moduli spaces of such embeddings and establishes their rigidity.

Findings

Moduli spaces of embeddings are explicitly described.

The real standard map is shown to be rigid.

Generalization of do Carmo--Wallach theorem to this context.

Abstract

We discuss holomorphic isometric embeddings of the projective line into quadrics using a generalisation of the theorem of do Carmo--Wallach to provide a description of their moduli spaces up to image and gauge--equivalence. Moreover, we show rigidity of the real standard map from the projective line into quadrics.