Minimal Norm Tensors And Its Two Applications In Geometry

In this paper we study the minimal norm tensors for general third covariant tensors.Using this we can give a new explanation of Cotton tensor: Cotton tensor is the minimal norm tensor of divergence of Riemannian curvature tensor,and we also get some useful inequalities by computation the norm of minimal norm tensors.As an application,we apply it to the symmetrization of the third-order covariant derivative of a smooth function on a compact Riemannian manifold,and then we get the generalized Lichnerowicz-Obata theorem.In particular,we prove that if the eigenvalue of Lapalacian on Einstein manifold with positive scalar curvature is greater than the first eigenvalue of the sphere（with the same scalar curvature）and smaller than the second eigenvalue of the sphere,then the eigenvalue can be controlled by the norm of the Weyl tensor.