Category:Isometries
This category contains results about Isometries.
Definitions specific to this category can be found in Definitions/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)
Subcategories
This category has the following 6 subcategories, out of 6 total.
I
- Isometries (Inner Product Spaces) (empty)