Cartan–Hadamard conjecture

The Cartan–Hadamard conjecture states that geodesic balls in a complete, simply connected Riemannian manifold of nonpositive sectional curvature have at least the volume of Euclidean balls with the same radius.

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Cartan–Hadamard theorem: A theorem stating that a complete, simply connected Riemannian manifold of nonpositive sectional curvature is diffeomorphic to Euclidean space via the exponential map. Its global exponential-map description supplies the setting for comparing geodesic balls with Euclidean balls.
Élie Cartan: A French mathematician whose work helped establish foundational results in differential geometry. His name is attached to the theorem underlying the conjecture's geometric setting.
Two-dimensional Cartan–Hadamard conjecture: The surface case of the Cartan–Hadamard conjecture, established through comparison of geodesic disks with Euclidean disks. The conjecture is known in dimension two, providing a settled low-dimensional case.
Cartan–Hadamard conjecture in higher dimensions: The volume inequality proposed for geodesic balls in Hadamard manifolds of dimension greater than two. The conjecture is unresolved in general beyond the established surface case.
Jacobi field: A vector field along a geodesic that describes the infinitesimal separation of nearby geodesics. Jacobi fields determine how curvature changes the radial volume density.
Jacques Hadamard: A French mathematician known for contributions to analysis, partial differential equations, and geometry. His name, alongside Cartan's, identifies the theorem whose setting frames the conjecture.
Three-dimensional Cartan–Hadamard conjecture: The restriction of the Cartan–Hadamard volume conjecture to complete, simply connected three-dimensional manifolds of nonpositive curvature. This dimension is a central case in the conjecture's unresolved higher-dimensional landscape.
Volume comparison theorem: A theorem bounding geometric volumes by those in a model space under curvature assumptions. The conjecture asks for a particularly strong lower comparison with the flat model.
Riemannian volume density: The local factor that converts coordinate volume into volume measured by a Riemannian metric. The conjectured inequality follows from integrating radial volume density over a ball.
Riemannian geometry: The study of smooth manifolds equipped with smoothly varying inner products on their tangent spaces. The conjecture is a global volume question within this geometric framework.