Immersion in $\mathbb{R}^n$ by complex spinors

TL;DR

This paper extends a spinor-based method for isometric immersions in Euclidean space to manifolds with SpinC-structures, broadening its applicability to complex manifolds.

Contribution

It adapts the existing spinor solution for isometric immersions to SpinC-structures, making it more suitable for complex manifolds.

Findings

Extension of spinor method to SpinC-structures

Broader applicability to complex manifolds

Maintains effectiveness of immersion solutions

Abstract

A beautiful solution to the problem of isometric immersions in $\mathbb{R}^n$ using spinors was found by Bayard, Lawn and Roth. However to use spinors one must assume that the manifold carries a $\mbox{Spin}$-structure and, especially for complex manifolds where is more natural to consider $\mbox{Spin}^{\mathbb{C}}$-structures, this hypothesis is somewhat restrictive. In the present work we show how the above solution can be adapted to $\mbox{Spin}^{\mathbb{C}}$-structures.