Abstract.
Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A g 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 {σ k (A g ), 1 ≤ k ≤ n} of the eigenvalues of A g with respect to g; we call σ k (A g ) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A g is semi-positive definite and σ k (A g ) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ2(A g ) 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.
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Zejun Hu: Partially supported by grants of DAAD, TU Berlin, Dierks von Zweck Stiftung at Essen, all in Germany; and SRF for ROCS, SEM; and grant No. 10671181 of NSFC
Haizhong Li: Partially supported by the Zhongdian grant No. 10531090 of NSFC.
Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC.
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Hu, Z., Li, H. & Simon, U. Schouten curvature functions on locally conformally flat Riemannian manifolds. J. geom. 88, 75–100 (2008). https://doi.org/10.1007/s00022-007-1958-z
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DOI: https://doi.org/10.1007/s00022-007-1958-z