We study the Obata equation with Robin boundary condition $\partial f/\partial \nu + af = 0$ on manifolds with boundary, where $a$ is a non-zero constant. Dirichlet and Neumann boundary conditions were previously studied by Reilly, Escobar and Xia. Compared with their results, the sign of $a$ plays an important role here. The new discovery shows besides spherical domains, there are other manifolds for both $a > 0$ and $a < 0$. We also consider the Obata equation with non-vanishing Neumann condition $\partial f/\partial \nu=1$.

中文翻译：

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具有Robin边界条件的Obata方程

我们研究了带边界流形上具有Robin边界条件$ \ partial f / \ partial \ nu + af = 0 $的Obata方程，其中$ a $是一个非零常数。赖利（Reilly），埃斯科巴尔（Escobar）和夏（Xia）先前曾研究Dirichlet和Neumann边界条件。与他们的结果相比，$ a $的符号在这里起着重要的作用。新发现表明，除了球形域外，对于$ a> 0 $和$ a <0 $还有其他流形。我们还考虑了具有不消失的诺伊曼条件$ \ partial f / \ partial \ nu = 1 $的Obata方程。