通过平面波极限研究几何
Geometry via Plane wave limits
AI总结：
本文利用协变表述为任意半黎曼度量构造等距不变的“平面波极限”，推广了 Penrose 极限，并借此将 Hawking & Penrose 的曲率结果部分推广至半黎曼情形，同时证明了 Morse 指数定理。
摘要：
利用 Blau 等人对 Penrose 平面波极限的协变表述，我们为任意半黎曼度量 $g$ 构造了一族“平面波极限”。这些极限沿 $g$ 的任意测地线取得，产生更简单的洛伦兹符号度量，并且是等距不变量。我们证明，它们将 Penrose 极限推广到半黎曼情形，并在某些情况下编码了 $g$ 的张量几何及其测地偏离。作为后者的应用，我们将 Hawking & Penrose 的一个著名结果部分推广到半黎曼情形：在任何半黎曼流形上，如果 Ricci 曲率沿着任何没有共轭点且“因果独立”（我们明确了其含义）的完备测地线非负，那么沿该测地线的曲率张量必须在所有法方向上消失。此外，还为这类测地线证明了 Morse 指数定理。
英文摘要：
Utilizing the covariant formulation of Penrose's plane wave limit by Blau et~al., we construct for any semi-Riemannian metric $g$ a family of "plane wave limits." These limits are taken along any geodesic of $g$, yield simpler metrics of Lorentzian signature, and are isometric invariants. We show that they generalize Penrose's limit to the semi-Riemannian regime and, in certain cases, encode $g$'s tensorial geometry and its geodesic deviation. As an application of the latter, we partially extend a well known result by Hawking & Penrose to the semi-Riemannian regime: On any semi-Riemannian manifold, if the Ricci curvature is nonnegative along any complete geodesic without conjugate points that is "causally independent" (in a sense we make precise), then the curvature tensor along that geodesic must vanish in all normal directions. A Morse Index Theorem is also proved for such geodesics.
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