Schouten curvature functions on locally conformally flat Riemannian manifolds |
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Authors: | Zejun Hu Haizhong Li Udo Simon |
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Affiliation: | 1. Department of Mathematics, Zhengzhou University, Zhengzhou, 450052, People’s Republic of China 2. Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, People’s Republic of China 3. Institut für Mathematik, MA 8-3, Technische Universit?t Berlin, Strasse des 17. Juni 136, 10623, Berlin, Germany
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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. 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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Keywords: | Primary 53C20 Secondary 53C25 |
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