The Bochner Formula for Riemannian Flows

On a Riemannian manifold ( M , g ) endowed with a Riemannian flow, we study in this paper the curvature term in the Bochner–Weitzenböck formula of the basic Laplacian. We prove that this term splits into two parts; a first part that depends on the curvature operator of the manifold M and a second part that can be expressed in terms of the O’Neill tensor of the flow. After getting a lower bound for this curvature term depending on a bound of each of these two parts, we establish an eigenvalue estimate for the basic Laplacian. We then discuss the limiting case of this latter estimate and prove that, when equality occurs, the manifold M is isometric to a local product.