Comparison theorem

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A comparison theorem is a result that relates, or compares, properties and quantities associated with a given Riemannian manifold, to properties and quantities associated with a Riemannian manifold equipped with a constant-curvature metric, viz a metric of constant sectional curvature.

The model Riemannian manifold (the one with constant-curvature) is typically taken to be a Riemannian manifold whose constant curvature is a bound on the sectional curvature (or Ricci curvature) of the given Riemannian manifold.