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 $\R^m.$
Categories: math.DG