Operator □～r On A Submanifold Of A Riemannian Manifold And Its Applications

Let V and W be n-dimensional vector space and p-dimensional vector space, respectively, let V* be dual space of V and V*(?)V*(?)W be a vector space. We denote a basis of V and a basis of W by {e_i}(i = 1, …, n), {e_α}(α = 1, … ,p), respectively. Let D = and D be a symmetric tensor whichmeans D_ij^α = D_ji^α, where ω_i is defined as dual basis of e_i. In this paper we first define the r-th Newton tensor T_(r)(D)(r = 0,1, … , n) which is determined by the tensor D of type (1,2), and we call it generalized Newton tensor; When V is a tangent space of a submanifod, and D is the second fundametal form of the submanifold (associated by the metric), the r-th elemetary symmetric functions are called the modified mean curvatures. Following this, in this paper we define " the r-th modified mean curvatures " of D_ij^α and call them as Q_r, we also study some algebraic properties of the r-th Newton tensor associated by the " r-th modified mean curvatures " and the properties of them for a submanifold of a space with constant sectional curvature. , and so these definitions and properties are natural generalization of classical Newton tensor and the r-th elemetary symmetric polynomial's definitons and properties for them(see[16]). Then, following the operator introduced by Cheng-Yau in [4] and by using these Newton tensor we induce a series of differential operators â–¡^r, which are adjoint relative to the L²â€”inner product. In the study of those properties, we find a new way to prove Minkowski-Hsiung integral formula. and derive some integral formulas for compact submanifolds,which are analogous to the Minkowski-Hsiung integral formula. Considering the case that â–¡^r acts on Q_r, we obtain two general conclusions. Finally we focus on the â–¡² operator for a hypersurface of a Riemannian manifold with harmonic Riemnnia curvature to study, and obtain a result of [20].