Abstract.
We show that if all geodesics of two non-proportional metrics on a closed manifold coincide (as unparameterized curves), then the manifold has a finite fundamental group or admits a local-product structure. This implies that, if the manifold admits a metric of negative sectional curvature, then two metrics on the manifold have the same geodesics if and only if they are proportional.
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Oblatum 18-IV-2002 & 12-VIII-2002¶Published online: 18 December 2002
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Matveev, V. Hyperbolic manifolds are geodesically rigid. Invent. math. 151, 579–609 (2003). https://doi.org/10.1007/s00222-002-0263-6
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DOI: https://doi.org/10.1007/s00222-002-0263-6