Isometric Homotopy in Codimension Two

Abstract

This article investigates the structure of the space of isometric immersions from a simply connected $n$-dimensional Riemannian manifold with positive sectional curvatures into $(n + 2)$-dimensional Euclidean space ${E^{n + 2}}$. It is proven that if $n \geqslant 4$ and ${M^n}$ is such a manifold which admits a ${C^\infty }$ isometric immersion as a hypersurface in ${E^{n + 1}}$, then any ${C^\infty }$ isometric immersion from ${M^n}$ into ${E^{n + 2}}$ is ${C^{2n - 4}}$ homotopic through isometric immersions to an immersion whose image lies in some hyperplane.