The Bochner Formula for Riemannian Flows

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