Hyperbolic manifolds are geodesically rigid
- Cite this article as:
- Matveev, V. Invent. math. (2003) 151: 579. doi:10.1007/s00222-002-0263-6
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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.