Isometric immersions of Riemannian manifolds in $k$-codimensional Euclidean space

TL;DR

This paper introduces a new method to determine when a Riemannian manifold can be locally isometrically immersed in Euclidean space of higher codimension, linking curvature conditions to scalar field equations.

Contribution

It develops a novel approach connecting local isometric immersions to scalar fields satisfying non-linear equations, generalizing classical results and providing new criteria for existence.

Findings

Conditions for local isometric immersion based on scalar fields.

Recovery of the fundamental theorem of hypersurfaces as a special case.

Reduction of Gauss and Codazzi equations to obstructions involving curvature logarithm.

Abstract

We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields on the manifold, satisfying a certain non-linear equation involving the Riemannian curvature tensor of $M$. Setting $k=1$, we proceed to recover the fundamental theorem of hypersurfaces. In the case of manifolds of positive sectional curvature and $n\geq 3$, we reduce the solvability of the Gauss and Codazzi equations to the cancelation of a set of obstructions involving the logarithm of the Riemann curvature operator. The resulting theorem has a structural similarity to the Weyl-Schouten theorem, suggesting a parallelism between conformally flat $n$-manifolds and those that admit an isometric immersion in $\mathbb{R}^{n+1}$.