Quaternion Kaehlerian Manifolds Isometrically Immersed in Euclidean Space

Abstract

Let $M$ be a complete $4n$-dimensional quaternion Kaehlerian manifold isometrically immersed in the $(4n + d)$-dimensional Euclidean space. In this note we prove that if $d < n$, then $M$ is a Riemannian product ${Q^m} \times P$, where ${Q^m}$ is the $m$-dimensional quaternion Euclidean space $(m \geqslant n - d)$ and $P$ is a Ricci flat quaternion Kaehlerian manifold.