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A well-known result asserts that any isometric immersion with flat normalbundle of a Riemannian manifold with constant sectional curvature into a spaceform is (at least locally) holonomic. In this note, we show that thisconclusion remains valid for the larger class of Einstein manifolds. As anapplication, when assuming that the index of relative nullity of the immersionis a positive constant we conclude that the submanifold has the structure of ageneralized cylinder over a submanifold with flat normal bundle.

Einstein submanifolds with flat normal bundle in space forms are holonomic