In this paper, we discuss inverse problem in spray geometry. We find infinitely many sprays with non-diagonalizable Riemann curvature on a Lie group, these sprays are not induced by Finsler metrics. We also study left invariant sprays with non-vanishing spray vectors on Lie groups. We prove that if such a spray S on a Lie group G satisfies that G is commutative or S is projective, then S is not induced by any (not necessary positive definite) left invariant Finsler metric. Finally, we construct an abundance of the left invariant sprays on Lie groups which satisfy the conditions in above result.

中文翻译：

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李群左不变喷射的逆问题

在本文中，我们讨论了喷雾几何中的逆问题。我们在李群上发现了无限多具有不可对角化黎曼曲率的喷雾，这些喷雾不是由 Finsler 度量引起的。我们还研究了李群上具有非消失喷雾向量的左不变喷雾。我们证明，如果这样的喷雾小号在李群上G满足G是可交换的或小号是射影的，那么小号不是由任何（不一定是正定的）左不变 Finsler 度量引起的。最后，我们在满足上述结果的条件的李群上构造了丰富的左不变喷雾。