Lichnerowicz--Obata first eigenvalue and round-sphere rigidity
Formalizes attainment of the first positive smooth Laplace eigenvalue on closed connected smooth Riemannian manifolds of dimension n >= 2, its sharp lower bound n*K under Ric >= (n-1)*K with K > 0, and equality if and only if the manifold is smoothly Riemannian-isometric to the round sphere of radius 1/sqrt K. The independent statement constructs the metric-compatible torsion-free connection and explicitly defines Ricci curvature, the Riesz gradient, the div-grad Laplacian, eigenvalue minimality, and the radius-scaled induced round metric.
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