Every Riemannian metric or Finsler metric on a manifold induces a spray via its geodesics. In this paper, we discuss several expressions for the χ-curvature of a spray. We show that the sprays obtained by a projective deformation using the S-curvature always have vanishing χ-curvature. Then we establish the Beltrami Theorem for sprays with χ = 0.

中文翻译：

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在 χ 曲率消失的喷雾上

歧管上的每个黎曼度量或 Finsler 度量都会通过其测地线引起喷雾。在本文中，我们讨论了几个表达式χ- 喷雾的曲率。我们表明，使用投影变形获得的喷雾小号-曲率总是消失χ-曲率。然后我们建立用于喷雾的贝尔特拉米定理χ = 0.