Isometry group ($\mathrm{Isom}(M,h)$)

The isometry group of a metric space consists of its bijective isometries, with composition as the group operation. For a Riemannian metric, these are the diffeomorphisms preserving the metric tensor. Right translations of a right-invariant Riemannian metric supply a faithful subgroup isomorphic to the underlying Lie group.

A finite group of orientation-preserving isometries of the Euclidean plane or hyperbolic plane is a cyclic group. Take one orbit and its unique minimum-radius enclosing disc. Every group element preserves the orbit and therefore the disc's centre. The orientation-preserving stabilizer of a point acts as planar rotations, so the group embeds in the circle group. In a nontrivial finite subgroup, the least positive rotation angle generates all angles by division with remainder, proving cyclicity. Orientation reversal changes the conclusion: reflection groups need not be cyclic.

For a connected three-dimensional Riemannian manifold, the Myers-Steenrod theorem bounds the dimension of its isometry group by six. Dimension five cannot occur: isotropy embeds in $O(3)$, whose Lie subalgebras have dimensions zero, one or three. A five-dimensional action would require three-dimensional isotropy and a two-dimensional orbit; full rotational isotropy has no invariant tangent two-plane. The maximal dimension six forces transitivity and isotropy, hence constant sectional curvature. Each of the six listed dimensions is attained by a suitable metric.

For a complete simply connected Riemannian manifold $X$ and a free properly discontinuous deck group $\Gamma$, each isometry of $X/\Gamma$ lifts to an isometry of $X$ normalizing $\Gamma$. Conversely such a normalizing isometry descends; two lifts differ by a deck transformation. This proves the formula, including orientation-reversing isometries if the full normalizer is used.

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 $O(n)$, giving dimension at most $n(n+1)/2$. For a compact manifold the isometry group is compact.

A finite group of Euclidean isometries has a common fixed point: average any orbit. Each isometry is an affine map and therefore preserves this finite affine average; applying a group element merely permutes the orbit terms. To see that an isometry is affine, subtract its value at zero and use the polarization identity to show that the resulting map preserves inner products. Images of a standard orthonormal basis then determine a linear orthogonal map.