Schouten Tensor On The Riemannian Manifold With Harmonic Conformal Curvature And Its Applications

In this paper, we study the Schouten tensor, which is expressed by the Ricci curvature and scalar curvature(cf[13]). We find that it is a Codazzi tensor on a Rie-mannian manifold M(dimM > 3)with harmonic Weyl conformal curvature tensor. Thus we regard it as a natural generalization of the second fundamental form of a hypersurface in real space form. Furthermore, we get some theorems and results. By using this tensor, we induce an operator â–¡, which is self-adjoint relative to the L₂- inner product. Then we characterize Einstein manifold and constant sectional curvature by inequalities between certain function on a compact local conformally symmetric space and a compact local conformally flat space respectively. Some new theorems are established and some known results are generalized.Our results can be stated as followsTheorem 2.1 Let M be a compact conformally symmetric space, dim(M) > 3. If M has constant scalar curvature and positive sectional curvature, then M is a Einstein manifold.Corollary 2.1 Let M be a compact conformally flat space, dim(M) > 3. If M has constant scalar curvature and positive sectional curvature, then M is of constant curvature space.When we weaken the condition of positive sectional curvature in Theorem2.1 to the condition of nonnegative sectional curvature , we obtain the theorem and corollary as follows.Theorem 2.2 Let M be a compact conformally symmetric space, dim(M) > 3. If M has constant scalar curvature and nonnegative sectional curvature, theneithor M is a Einstein manifold,orM is the product of some Riemannian manifolds , i.e. M = Mⁿi Mⁿr(n₁ + n_r = dimM), every Mⁿi(l < i < r) is Einstein manifold.Corollary 2.2 Let M be a compact conformally symmetric space,dim(M) > 3. If M has constant scalar curvature and nonnegative sectional curvature, then M is the product of two Riemannian manifold with constant sectional curvature , i.e. M = M^p{c) x Mn-p(-c).Theorem 2.3 Let M be a compact conformally symmetric space with positive sectional curvature , ifthen M is Einstein.Corollary 2.3 Let M(dimM > 3) be a compact conformaUy flat space with positive sectional curvature. Ifthen M is a constant sectional curvature space.Theorem 3.1 Let M(dimM > 3) be a compact conformaUy flat manifold. If S satisfiesThen, eitherand M is constant sectional curvature space; orand S has two different eigenvalues Ai, An,where1,. 2nCorollary 3.1 Let M(dimM > 3) be a compact conformally flat manifold withconstant scalar curvature r. IftrS>then M is constant sectional curvature space.We Generalize the Goldberg-Okumur a's pinching theorem and obtain Corollary 3.2 Let M(dimM > 3) be a compact conformaUy flat manifold. If(OEij.fc.i J%fci A = c(= const.),(ii)IIQIl < ^r-then M is constant sectional curvature space.