A version of the theorem of Johnson,Palmer, and Sell for quasicompact cocycles

We provide a classification of Einstein submanifolds in space forms with flat normal bundle and parallel mean curvature. This extends a previous result due to Dajczer and Tojeiro (Tohoku Math J (2) 45:43–49, 1993) for isometric immersions of Riemannian manifolds with constant sectional curvature.