The Bochner formula for isometric immersions

We study the Bochner formula for a manifold isometrically immersed into another and obtain a Gauss-type splitting of its curvature term.In fact, we prove that the curvature term in the Bochner formula is an operator that can be explicitly expressed in terms of the curvature operator of the ambient manifold and the extrinsic geometry (second fundamental form) of the immersion.Several applications of this splitting are given, namely, eigenvalue estimates for the Hodge Laplacian, vanishing results for the de Rham cohomology and rigidity of immersions of Kähler manifolds into negatively curved spaces.