Myers's theorem is a result in Riemannian geometry, which concerns the relationship between the geometry of a complete Riemannian manifold and its topology. Specifically, the theorem states that if \( M \) is a complete Riemannian manifold that has non-negative Ricci curvature, then \( M \) can be isometrically embedded into a Euclidean space of a certain dimension.
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For a connected Riemannian manifold of dimension , assume geodesic completeness and for a constant . The Riemannian distance then has diameter at most ; the manifold is compact and its fundamental group is finite. The dimension restriction excludes the vacuous one-dimensional Ricci bound.