We give the full classification of left-invariant metrics of an arbitrary signature on the Lie group corresponding to the real hyperbolic space. It is shown that all metrics have constant sectional curvature and that they are geodesically complete only in the Riemannian case.

中文翻译：

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ℝH n上左不变度量的测地完备性

我们给出了与实际双曲空间相对应的李群上任意签名的左不变度量的完整分类。结果表明，所有度量均具有恒定的截面曲率，并且仅在黎曼情形下它们才是测地学上完整的。