Manifolds with harmonic Weyl curvature and curvature operator of the second kind

We prove that a compact Riemannian manifold of dimension $n\ge 8$ with harmonic Weyl curvature and $\frac{3(n-1)(n+2)}{4(3n-1)}$-nonnegative curvature operator of the second kind is either globally conformally equivalent to a space of positive constant curvature or is isometric to a flat manifold. In particular, We also give a classification of four-dimensional manifolds with harmonic Weyl curvature satisfying a cone condition. This result generalizes the work in \cite{DFY24,FLD,Li22}.

💡 Research Summary

The paper investigates the interplay between harmonic Weyl curvature and the curvature operator of the second kind on compact Riemannian manifolds. The authors focus on manifolds of dimension (n\ge 8) whose Weyl tensor is divergence‑free ((\delta W=0)) and whose second‑kind curvature operator (\mathring R) satisfies a quantitative non‑negativity condition that is stronger than previously considered cone conditions. Specifically, they require that (\mathring R) be (\displaystyle k_0)-non‑negative with
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