Spinorial Representation of Submanifolds in Riemannian Space Forms

TL;DR

This paper introduces a spinorial method to represent submanifolds in Riemannian space forms using generalized Killing spinors, providing new insights and proofs in submanifold theory.

Contribution

It presents a novel spinorial representation for submanifolds of any dimension and codimension, unifying and extending previous results in the field.

Findings

Provides a new proof of the fundamental theorem of submanifold theory

Recovers known results in dimensions 2 and 3

Establishes a framework for representing submanifolds via spinors

Abstract

In this paper we give a spinorial representation of submanifolds of any dimension and codimension into Riemannian space forms in terms of the existence of so called generalized Killing spinors. We then discuss several applications, among them a new and concise proof of the fundamental theorem of submanifold theory. We also recover results of T. Friedrich, B. Morel and the authors in dimension 2 and 3.