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In this article we make a local classification of $n$-dimensional Riemannian manifolds $(M,g)$ with harmonic curvature and less than four Ricci eigenvalues which admit a smooth non constant solution $f$ to the following equation \begin{align} \label{0002bxu} \nabla df = f(r -\frac{R}{n-1} g) + x \cdot r+ y(R) g, \end{align} where $\nabla $ is the Levi-Civita connection of $g$, $r$ is the Ricci tensor of $g$, $x$ is a constant and $y(R)$ a function of the scalar curvature $R$. Indeed, we showed that, in a neighborhood $V$ of each point in some open dense subset of $M$, either {\rm (i)} or {\rm (ii)} below holds; {\rm (i)} $(V, g, f+x)$ is a static space and isometric to a domain in the Riemannian product of an Einstein manifold $N$ and a static space $(W, g_W, f+x)$, where $g_W$ is a warped product metric of an interval and an Einstein manifold. {\rm (ii)} $(V, g)$ is isometric to a domain in the warped product of an interval and an Einstein manifold. For the proof we use eigenvalue analysis based on the Codazzi tensor properties of the Ricci tensor.
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참고문헌 (18)
참고문헌 신청
L. Ambrozio / 2017 / On static three-manifolds with positive scalar curvature / J. Dierential Geom 107(1):1~45
H. Baltazar / On the volume functional of compact manifolds with boundary with harmonic Weyl tensor / arXiv:1710.06247v1 [math.DG]
H. Baltazar / 2018 / Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary / Pacic J. Math 297(1):29~45
A. Barros / 2014 / Critical point equation on four-dimensional compact manifolds / Math. Nachr 287(14-15):1618~1623
R. Batista / 2017 / Critical metrics of the volume functional on compact three-manifolds with smooth boundary / J. Geom. Anal 27(2):1530~1547
H. Baltazar / On the volume functional of compact manifolds with boundary with harmonic Weyl tensor / arXiv:1710.06247v1 [math.DG]
H. Baltazar / 2018 / Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary / Pacic J. Math 297(1):29~45
A. Barros / 2014 / Critical point equation on four-dimensional compact manifolds / Math. Nachr 287(14-15):1618~1623
R. Batista / 2017 / Critical metrics of the volume functional on compact three-manifolds with smooth boundary / J. Geom. Anal 27(2):1530~1547
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