Distances are often more important than coordinates. A rotation of the plane changes every coordinate pair, but it does not change the shape of any figure; translating a function in a Hilbert space may move it to a different location, but the norm can remain unchanged. An isometry is the abstraction of this exact preservation. It is the map that lets geometry be transported without distortion, whether the ambient object is a [metric space](/page/Metric%20Space), an [inner product space](/page/Inner%20Product%20Space), a [Hilbert space](/page/Hilbert%20Space), or a [Riemannian manifold](/page/Cambridge%20III%20Riemannian%20Geometry).

The word has several closely related uses. In metric geometry it means preservation of distance. In linear analysis it often means preservation of norm, especially for bounded operators on normed or Hilbert spaces. In finite-dimensional [inner product](/page/Inner%20Product) geometry it means preservation of the inner product itself, giving the familiar orthogonal and unitary maps. In Riemannian geometry it means preservation of the metric tensor through a diffeomorphism. These are not competing ideas: they are manifestations of the same principle at different levels of structure.

## Definition

The most flexible setting is a metric space, where the only primitive geometric data is the distance between two points. A map deserves to be called rigid in this setting if every pairwise distance is preserved, since all metric statements about balls, convergence, diameter, and Cauchy behaviour are encoded by those distances.

[definition: Isometry] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A map $f:X\to Y$ is an isometry if \begin{align*} d_Y(f(x),f(y)) = d_X(x,y) \end{align*} for all $x,y\in X$. [/definition]

This definition does not require $f$ to be onto. Many authors reserve the word isometry for the surjective case and call the general case an isometric embedding. When a page needs to stress that the source sits inside the target as an undistorted copy, the embedding terminology makes that role explicit.

[definition: Isometric Embedding] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. An isometric embedding is an isometry $f:X\to Y$. [/definition]

A map that preserves all distances cannot identify two distinct points, since the distance between distinct points is positive. Sometimes, however, the goal is not to place $X$ inside $Y$, but to identify $X$ and $Y$ as the same metric object with different labels. That requires a distance-preserving map that reaches every point of the target.

[definition: Surjective Isometry] Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces. A surjective isometry is an isometry $f:X\to Y$ such that for every $y\in Y$ there exists $x\in X$ with $f(x)=y$. [/definition]

The surjective condition is what turns an undistorted copy into a genuine identification of metric spaces. For this identification to be usable, it must work in both directions: after relabelling $X$ as $Y$, we should be able to recover $X$ from $Y$ without introducing distortion. The next result records that no extra hypothesis is needed for this reverse map; surjectivity and distance preservation already force the inverse to preserve distance.

[quotetheorem:8698]

This result justifies treating surjective isometries as genuine sameness of metric spaces rather than merely as special maps between them. The surjectivity hypothesis is essential: a non-surjective isometric embedding, such as the inclusion of a closed interval into a larger interval, preserves every distance in its domain but has no inverse defined on the whole target. In that example, distances inside the smaller interval are unchanged, but points of the larger interval outside the image have no preimage at all, so statements about the whole target cannot be transported back.

The theorem also explains why the inverse map requires no separate continuity or Lipschitz argument. Once every point of the target has a unique preimage, the equation defining distance preservation can be read in reverse, so metric balls, convergent sequences, Cauchy behavior, diameters, and other distance-defined data transfer back to the source. This is the practical reason the surjective condition appears throughout later invariance statements: it lets one replace a metric space by an isometric model without changing what is true about the whole space.

Bijectivity alone would not be enough either, since an arbitrary bijection can stretch distances; the point is that the inverse inherits distance preservation from the original map. This is why surjective isometries are the maps that allow metric arguments, examples, and invariants to be transported back and forth without changing their content. The same observation also explains the categorical role of these maps: they are precisely the morphisms that can be undone within the metric category.

[remark: Categorical Role of Surjective Isometries] In the category whose objects are metric spaces and whose morphisms are distance-preserving maps, the isomorphisms are precisely the surjective isometries. [/remark]

This categorical viewpoint is useful, but many analytic settings still need non-surjective isometries, especially when one space embeds into a larger one.

In normed vector spaces, distance is usually generated by a norm. For a linear map, preserving all distances is equivalent to preserving the norm of every vector. This is why the operator-theoretic definition is often written using only $\|Tx\|$.

[definition: Linear Isometry] Let $X$ and $Y$ be normed vector spaces over $\mathbb R$ or $\mathbb C$, with norms $\|\cdot\|_X$ and $\|\cdot\|_Y$. A linear map $T:X\to Y$ is a linear isometry if \begin{align*} \|Tx\|_Y = \|x\|_X \end{align*} for all $x\in X$. [/definition]

Hilbert spaces are the central normed spaces of analysis because their norms come from inner products. To study shifts, partial symmetries, and unitary dynamics, we need a version of isometry for bounded operators that preserves Hilbert norm while allowing the range to be a proper closed subspace.

[definition: Hilbert Space Isometry] Let $H$ and $K$ be Hilbert spaces over $\mathbb R$ or $\mathbb C$, and let $\mathcal{L}(H,K)$ denote the space of bounded linear maps from $H$ to $K$. An operator $V\in \mathcal{L}(H,K)$ is a Hilbert space isometry if \begin{align*} \|Vx\|_K = \|x\|_H \end{align*} for all $x\in H$. [/definition]