Schouten curvature functions on locally conformally flat Riemannian manifolds

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor Ag associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σk(Ag), 1 ≤ k ≤ n} of the eigenvalues of Ag with respect to g; we call σk(Ag) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: Ag is semi-positive definite and σk(Ag) is a nonzero constant for some k ∈ {2, · · · , n}. If k = 2, we obtain a classification result under the weaker conditions that σ2(Ag) is a non-negative constant and (M n, g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature. Mathematics Subject Classification (2000): Primary 53C20; Secondary 53C25.