Cartan–Hadamard conjecture

In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and Geometric measure theory which states that the classical isoperimetric inequality may be generalized to spaces of nonpositive sectional curvature, known as Cartan–Hadamard manifolds. The conjecture, which is named after French mathematicians Élie Cartan and Jacques Hadamard, may be traced back to work of André Weil in 1926.

Informally, the conjecture states that negative curvature allows regions with a given perimeter to hold more volume. This phenomenon manifests itself in nature through corrugations on coral reefs, or ripples on a petunia flower, which form some of the simplest examples of non-positively curved spaces.