We revisit and generalize a recent result of Cederbaum (Cederbaum in preparation; Cederbaum 2017 Oberwolfach Report) concerning the rigidity of the Schwarzschild manifold for spin manifolds. This includes the classical black hole uniqueness theorems (Bunting and Masood-ul-Alam 1987 Gen. Relativ. Gravit. 19 147; Gibbons, Ida and Shiromizu 2002 Phys. Rev. Lett. 89 041101; Hwang 1998 Geom. Dedicata 71 5) as well as the more recent uniqueness theorems for photon spheres (Cederbaum 2015 Contemp. Math. 667 86; Cederbaum and Galloway 2017 Commun. Anal. Geom. 25 303; Cederbaum and Fehrenbach personal communication).

我们重新审视并概括了 Cederbaum（Cederbaum 准备中；Cederbaum 2017 Oberwolfach 报告）关于自旋流形的 Schwarzschild 流形刚性的最近结果。这包括经典的黑洞唯一性定理 (Bunting and Masood-ul-Alam 1987 Gen. Relativ. Gravit. 19 147; Gibbons, Ida and Shiromizu 2002 Phys. Rev. Lett. 89 041101; Hwang 1998 Geom. Dedicata ) as 71以及最近的光子球唯一性定理（Cederbaum 2015 Contemp . Math. 667 86；Cederbaum and Galloway 2017 Commun. Anal. Geom. 25 303；Cederbaum 和 Fehrenbach 个人交流）。