Conformal Kaehler Euclidean submanifolds

TL;DR

This paper classifies conformal Kaehler Euclidean submanifolds of certain codimensions, showing they can be locally constructed from isometric immersions into lower-dimensional Euclidean spaces, extending understanding of their geometric structure.

Contribution

It demonstrates that conformal Kaehler submanifolds with codimensions 1 and 2 are locally derived from isometric immersions into lower-dimensional Euclidean spaces, simplifying their classification.

Findings

Submanifolds with codimension 1 or 2 are locally obtainable from isometric immersions.

Such submanifolds are free of flat points and of a specific geometric type.

The results extend known classifications of Kaehler submanifolds in Euclidean spaces.

Abstract

Let $f\colon M^{2n}\to\mathbb{R}^{2n+\ell}$, $n \geq 5$, denote a conformal immersion into Euclidean space with codimension $\ell$ of a Kaehler manifold of complex dimension $n$ and free of flat points. For codimensions $\ell=1,2$ we show that such a submanifold can always be locally obtained in a rather simple way, namely, from an isometric immersion of the Kaehler manifold $M^{2n}$ into either $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n+2}$, the latter being a class of submanifolds already extensively studied.