标题： 齐次超空间上的爱因斯坦度量 显示英文标题

标题： Einstein metrics on homogeneous superspaces

摘要： 本文开创性地研究了齐次超流形上的爱因斯坦方程。 首先，我们为这些空间上分级黎曼度量显式地推导出曲率公式。 接下来，我们通过杜尔金图构造齐次超流形，这种方法类似于经典（非超）理论中广义旗流形的构造。 我们描述了通过此方法获得的几类空间上的爱因斯坦度量。 我们的结果提供了紧致齐次超流形上的例子，在这些流形上爱因斯坦方程无解、存在离散族解以及保度量下的连续族里奇平坦解。 这些例子表明，经典齐次几何中的有限性猜想在超流形上失效，并挑战了博赫纳消失定理所提供的直觉。 显示英文摘要

摘要： This paper initiates the study of the Einstein equation on homogeneous supermanifolds. First, we produce explicit curvature formulas for graded Riemannian metrics on these spaces. Next, we present a construction of homogeneous supermanifolds by means of Dynkin diagrams, resembling the construction of generalised flag manifolds in classical (non-super) theory. We describe the Einstein metrics on several classes of spaces obtained through this approach. Our results provide examples of compact homogeneous supermanifolds on which the Einstein equation has no solutions, discrete families of solutions, and continuous families of Ricci-flat solutions among invariant metrics. These examples demonstrate that the finiteness conjecture from classical homogeneous geometry fails on supermanifolds, and challenge the intuition furnished by Bochner's vanishing theorem.