Myers's theorem

Myers's theorem states that a complete Riemannian manifold with Ricci curvature bounded below by a positive constant is compact, has bounded diameter, and has finite fundamental group.

Bonnet–Myers theorem: A comparison theorem bounding the diameter of a complete Riemannian manifold with a positive Ricci curvature lower bound. This is the standard name for the diameter and compactness conclusion associated with Myers's theorem.
Riemannian manifold: A smooth manifold equipped with an inner product on each tangent space that varies smoothly from point to point. The theorem concerns global geometry induced by this smoothly varying metric.
Cartan–Hadamard theorem: A theorem stating that a complete simply connected manifold with nonpositive sectional curvature is diffeomorphic to Euclidean space. Its nonpositive-curvature setting leads toward noncompactness rather than Myers's compactness conclusion.
Diameter bound: An upper bound on the supremum of distances between pairs of points in a metric space. This is the quantitative geometric conclusion that drives compactness.
Second variation of arc length: A formula describing how the length of a geodesic changes under a variation of that geodesic. Test variation fields in this formula force minimizing geodesics to stop being minimizing beyond a curvature-controlled length.
Ricci curvature: A tensor that measures, in each direction, an averaged curvature of planes containing that direction. The theorem requires a uniform positive lower bound on this curvature.
Bonnet–Myers theorem with nonnegative Ricci curvature: A proposed theorem asserting compactness from a nonnegative Ricci curvature lower bound alone. Nonnegative Ricci curvature alone does not imply compactness, unlike the strictly positive lower bound in Myers's theorem.
Compact manifold: A manifold that is compact as a topological space. Completeness and the diameter bound together yield compactness in the theorem.
Index form: A quadratic form on vector fields along a geodesic that gives the second variation of its energy. Its nonnegativity for minimizing geodesics yields the contradiction behind the length bound.
Geodesic completeness: The property that every geodesic can be extended for all real values of its affine parameter. Completeness ensures the minimizing-geodesic arguments apply globally.

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Hopf conjecture (Euler characteristic)Compared with: Positive Ricci curvature gives strong global conclusions, but is weaker than positive sectional curvature.