Category:Definitions/Isometries

This category contains definitions related to Isometries.
Related results can be found in Category:Isometries.

Isometry (Metric Spaces)

Let $M_1 = \tuple {A_1, d_1}$ and $M_2 = \tuple {A_2, d_2}$ be metric spaces or pseudometric spaces.

Let $\phi: A_1 \to A_2$ be a bijection such that:

$\forall a, b \in A_1: \map {d_1} {a, b} = \map {d_2} {\map \phi a, \map \phi b}$

Then $\phi$ is called an isometry.

That is, an isometry is a distance-preserving bijection.

Isometry (Euclidean Geometry)

Let $\EE$ be a real Euclidean space.

Let $\phi: \EE \to \EE$ be a bijection such that:

$\forall P, Q \in \EE: PQ = P'Q'$

where:

$P$ and $Q$ are arbitrary points in $\EE$

$P'$ and $Q'$ are the images of $P$ and $Q$ respectively

$PQ$ and $P'Q'$ denote the lengths of the straight line segments $PQ$ and $P'Q'$ respectively.

Then $\phi$ is an isometry.

Isometry (Inner Product Spaces)

Let $V$ and $W$ be inner product spaces.

Let their inner products be $\innerprod \cdot \cdot_V$ and $\innerprod \cdot \cdot_W$ respectively.

Let the mapping $F : V \to W$ be a vector space isomorphism that preserves inner products:

$\forall v_1, v_2 \in V : \innerprod {v_1} {v_2}_V = \innerprod {\map F {v_1}} {\map F {v_2}}_W$

Then $F$ is called a (linear) isometry.

Isometry (Riemannian Manifolds)

Let $\struct {M, g}$ and $\struct {\tilde M, \tilde g}$ be Riemannian manifolds with Riemannian metrics $g$ and $\tilde g$ respectively.

Let the mapping $\phi : M \to \tilde M$ be a diffeomorphism such that:

$\phi^* \tilde g = g$

Then $\phi$ is called an isometry from $\struct {M, g}$ to $\struct {\tilde M, \tilde g}$.

Isometry (Bilinear Spaces)

Definition:Isometry (Bilinear Spaces)

Isometry (Quadratic Spaces)

Definition:Isometry (Quadratic Spaces)