The isometry group of a smooth connected Riemannian manifold is a finite-dimensional Lie group, and its action is smooth. An isometry is determined by its value and derivative at one point, since it preserves the exponential map and continuation along geodesics. The stabilizer embeds in , giving dimension at most . For a compact manifold the isometry group is compact.
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The Myers–Steenrod theorem is an important result in differential geometry, particularly in the study of Riemannian manifolds. It primarily deals with the structure of Riemannian manifolds that have certain properties related to curvature.