The paper reveals that isometries in hyperbolic space preserve horospheres and equidistant surfaces, highlighting their unique mathematical properties.
Let H n {H^n} be the n n -dimensional hyperbolic space. In this paper, we prove the following results: (i) An injective map f : H n ↦ H n f: {H^n}↦ {H^n} is an isometry if and only if it preserves horospheres. (ii) An injective map f : H n ↦ H n f:{H^n}↦ {H^n} is an isometry if and only if it preserves r r -equidistant surfaces. We do not assume continuity in these results.
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Gao et al. (2025) studied this question.
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