  Consider a closed manifold $M$ immersed in $\R^m.$ Suppose that the trivial
bundle $M\times\R^m=TM\otimes \nu M$ is equipped with an almost metric
connection $\tilde{\nabla}$ which almost preserves the decomposition of
$M\times\R^m$ into the tangent and the normal bundle. Assume moreover that the
difference $\Gamma=\partial-\tilde{\nabla}$ with the usual derivative
$\partial$ in $\R^m$ is almost $\tilde{\nabla}$-parallel. We show that under
these assumptions $M$ admits an extrinsically homogeneous immersion into
$\R^m.$
