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A smooth nondegenerate component of the fixed-point set of an isometry is a totally geodesic submanifold. If a geodesic starts tangent to the fixed component, its image under the isometry has identical position and tangent. Uniqueness of the geodesic equation makes the two geodesics coincide throughout their common domain, so the original geodesic stays in the fixed set.

If distinct points $x,y$ of a dynamical system on a compact metric space satisfy $d(T^{n}x,T^{n}y)\to 0$ in one compatible metric, no compatible metric can make $T$ an isometry. On a compact space, all compatible metrics give the same asymptotic-pair property by uniform continuity, whereas an isometry preserves the strictly positive distance between distinct points. In a full shift, a constant sequence and a sequence differing at just one coordinate converge to each other under forward shifts, proving this obstruction directly in the product topology.

A surjective linear isometry $U:C(K)\to C(L)$ between compact Hausdorff function spaces has the form $(Uf)(s)=u(s)f(h(s))$, where $h:L\to K$ is a homeomorphism and $u$ is continuous with modulus one. Thus the Banach space structure of $C(K)$ determines $K$ up to homeomorphism.

Two horocycles have a unique common orthogonal hyperbolic line when their ideal centres of horocycles differ: it is the line with those two ideal endpoints. If the centres agree, an isometry puts both horocycles on horizontal lines; every vertical hyperbolic line meets both orthogonally, so there are infinitely many. This classification remains valid for intersecting or tangent horocycles.

A C-star homomorphism is a complex linear multiplicative map between C-star algebras that preserves the involution. Such maps are contractive; an injective one is an isometry.

A finite point set is cyclic transitive when a cyclic group of isometries acts transitively on it.

For a probability measure-preserving system, if $⟨U_T^{n}f,f⟩\to |\int fd\mu {|}^2$, then $U_T^{n}f$ converges weakly in $L^2$ to $(\int fd\mu )1$. Centre $v=f-(\int f)1$; its autocorrelations tend to zero. The isometry identity gives the same decay against each $U_T^{k}v$, hence by approximation against their closed forward span. Against its orthogonal complement the correlations are exactly zero. Orthogonal decomposition by a closed subspace finishes the proof without invertibility.

Transforming a metric tensor and all matter fields together by the pullback of a diffeomorphism preserves the Einstein field equations and covariant matter equations. Passive coordinate changes only change the description of the fields; active pullbacks produce another representative of the geometry. The treatment of transformations acting on a boundary depends on the boundary conditions and possible charges. A diffeomorphism is not necessarily an isometry of a fixed metric.

For a complete invertible Hamiltonian flow $\Phi (t,s)$, integrate the source along the backward characteristic ending at $z$ at time $t$. The displayed formula solves the Hamiltonian Liouville equation. Volume preservation makes each composition with $\Phi (s,t)$ an isometry of Lp spaces.

A finite set $X\subseteq {\mathbb{R}}^d$ is Euclidean Ramsey when, for every number of colors, some finite-dimensional Euclidean space has the property that every coloring of it contains a monochromatic isometric copy of $X$.

This Hamming distance on partitions identifies a group with its complement because exchanging the two group names does not change the partition. Complementation is an isometry of the ordinary Hamming distance, so minimizing over this two-element group action gives a quotient metric and preserves the triangle inequality. Negating a zero-one indicator is not complementation; with $\pm 1$ membership vectors, it is.

A Riemannian manifold is homogeneous when its isometry group acts transitively on points. For example, a Lie group with a left-invariant metric is homogeneous under left translations. Homogeneity gives the same local metric-ball geometry at every point, which yields the completeness of homogeneous Riemannian manifolds. It does not imply two-point homogeneity: that stronger property also controls directions and pairs at equal distance.

A hyperbolic disc consists of points within a specified hyperbolic distance of its centre. In the Poincare disc model, the disc centred at zero with radius $\rho$ has Euclidean radius $\tanh (\rho /2)$; an isometry transports this description to any other centre.

Three corresponding side lengths determine a hyperbolic triangle up to an isometry of the hyperbolic plane. First match an ordered pair of vertices by an isometry. The third vertex lies at the intersection of two prescribed hyperbolic distance circles. There are at most two choices, interchanged by reflection in the geodesic through the matched pair; either choice can therefore be matched.

The ideal centre of a horocycle is its point of tangency to the boundary at infinity; a horizontal horocycle in the upper half-plane model has centre infinity. Sending the centre to infinity by an isometry makes the horocycle horizontal. Orthogonal hyperbolic lines then become vertical, proving that a hyperbolic line meets a horocycle orthogonally exactly when it has the centre as an ideal endpoint.

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.

The map $A$ associated with an affine map $x\mapsto Ax+b$; it is independent of the translation $b$. An affine Euclidean isometry has an orthogonal linear part, and an isometry between flat tori has a lift whose linear part carries one Euclidean lattice onto the other.

For a nonnegative smooth bump function supported in an open set, multiplying a Riemannian metric by $1+t\eta$ increases volume there for $t>0$ while leaving the metric elsewhere fixed. For distinct regular compact domains a perturbation supported in the interior of one outside the other breaks equal volume, and therefore breaks any isometry between the domains. The perturbations tend to the original metric in the smooth topology.