Geometric Rigidity of Manifolds Admitting m-Modified Conformal Vector Fields

We establish certain integral inequalities for m-modified conformal vector fields on compact Riemannian manifolds and derive corresponding triviality and rigidity results. We further obtain integrability conditions for such vector fields and establish an extended Bourguignon-Ezin identity. We then prove that a compact Riemannian manifold admitting a non-trivial closed m-modified conformal vector field is isometric to a round sphere under either a constant scalar curvature and positive Ricci curvature assumption or a suitable bound on the Ricci curvature involving the first nonzero eigenvalue of the Laplace operator. As a consequence, we obtain an extension of a classical sphere characterization result of Yano and Nagano to the corresponding result for a non trivial m-modified conformal vector field assumed closed. Finally, we consider a complete Kähler manifold admitting a closed m-modified conformal vector field V, and show that V is m-modified Killing and around a non-zero point of V the manifold splits as the product of the complex Euclidean plane C and a complete Kähler manifold.