Isometry

In mathematics, an isometry is a mapping that maps two metric spaces onto one another and thereby preserves the metric (distance, distance). This means that the distance between two image points is the same as that of the original image points.

In Euclidean and synthetic geometry , isometries are specifically considered that are also geometrical images for the spaces under consideration. Usually one speaks of a distance-maintaining , length- true or isometric image. If the required additional properties are clear from the context, simply from an isometry .

In contrast to this, in Riemannian geometry , an isometry is understood to be a mapping that contains the Riemannian metric and thus only the lengths of vectors and the lengths of curves. Such a mapping does not need to preserve the distances between two points.

definition

Are two metric spaces , given, and is a mapping with the property ${\ displaystyle (M_ {1}, d_ {1})}$ ${\ displaystyle (M_ {2}, d_ {2})}$ ${\ displaystyle f \ colon M_ {1} \ rightarrow M_ {2}}$

{\ displaystyle d_ {2} \ left (f (x), f (y) \ right) = d_ {1} (x, y) \}

for all ,

{\ displaystyle x, y \ in M_ {1}}

then isometry is called from to . Such a mapping is always injective . If it is even bijective , then it is called isometric isomorphism , and the spaces and are called isometric isomorphic; otherwise one calls an isometric embedding of in . ${\ displaystyle f}$ ${\ displaystyle M_ {1}}$ ${\ displaystyle M_ {2}}$ ${\ displaystyle f}$ ${\ displaystyle f}$ ${\ displaystyle M_ {1}}$ ${\ displaystyle M_ {2}}$ ${\ displaystyle f}$ ${\ displaystyle M_ {1}}$ ${\ displaystyle M_ {2}}$

Special cases

Normalized vector spaces

In normalized vector spaces , the distance between two vectors is defined by the norm of the difference vector: ${\ displaystyle V}$ ${\ displaystyle u, v \ in V}$

{\ displaystyle d (u, v) = \ | vu \ |}

.

If and are two normalized vector spaces with norm or and is a linear mapping , then this mapping is an isometry if and only if it receives the norm, i.e. if for all ${\ displaystyle V}$ ${\ displaystyle W}$ ${\ displaystyle \ | \ cdot \ | _ {V}}$ ${\ displaystyle \ | \ cdot \ | _ {W}}$ ${\ displaystyle f \ colon V \ to W}$ ${\ displaystyle v \ in V}$

{\ displaystyle \ | f (v) \ | _ {W} = \ | v \ | _ {V}}

applies.

Without the requirement of linearity, the following applies to real normalized vector spaces:

If the norm of the target space is strictly convex, each isometric by an affine transformation . ${\ displaystyle \ | \ cdot \ | _ {W}}$ ${\ displaystyle V}$ ${\ displaystyle W}$

Every surjective isometry is an affine mapping ( Mazur-Ulam theorem ).

In both cases the following applies: If the mapping maps the zero vector of onto the zero vector of , then it is linear. ${\ displaystyle V}$ ${\ displaystyle W}$

Vector spaces with dot product

If a vector space has a scalar product , then the induced norm (length) of a vector is defined as the square root of the scalar product of the vector with itself. For the distance between two vectors and we then get: ${\ displaystyle V}$ ${\ displaystyle u}$ ${\ displaystyle v}$

{\ displaystyle d (u, v) = \ | vu \ | = {\ sqrt {\ langle uv, uv \ rangle}}}

,

where the scalar product is denoted here by angle brackets.

If and are vector spaces with scalar product or and are linear mapping, then this mapping is linear isometry if and only if it receives the scalar product, that is ${\ displaystyle V}$ ${\ displaystyle W}$ ${\ displaystyle \ langle \ cdot, \ cdot \ rangle _ {V}}$ ${\ displaystyle \ langle \ cdot, \ cdot \ rangle _ {W}}$ ${\ displaystyle f \ colon V \ to W}$

{\ displaystyle \ langle f (u), f (v) \ rangle _ {W} = \ langle u, v \ rangle _ {V}}

for everyone .

{\ displaystyle u, v \ in V}

Such mappings are also called orthogonal mappings (in the case of real scalar product spaces) or unitary mappings (in the case of complex scalar product spaces). In the case of real scalar product spaces, it does not have to be assumed that the mapping is linear, because every isometry that maps the zero vector to the zero vector is linear in this case.

Is an orthonormal basis of , then a linear map if and only an isometric if one orthonormal system in is. ${\ displaystyle \ {a_ {1}, \ ldots, a_ {n} \}}$ ${\ displaystyle V}$ ${\ displaystyle f \ colon V \ to W}$ ${\ displaystyle \ {f (a_ {1}), \ ldots, f (a_ {n}) \}}$ ${\ displaystyle W}$

The set of all linear isometries of a Euclidean vector space forms a group , the orthogonal group of the space. Correspondingly, the set of all linear isometries of a unitary vector space forms the unitary group of the space.

Euclidean point space

Every isometry between two Euclidean point spaces and is an affine map . You can get in the shape ${\ displaystyle f \ colon E \ to F}$ ${\ displaystyle E}$ ${\ displaystyle F}$

{\ displaystyle f (Q) = f (P) + {\ vec {f}} ({\ overrightarrow {PQ}})}

for all

{\ displaystyle P, Q \ in E}

represent, where a linear isometry between the associated Euclidean vector spaces and is. ${\ displaystyle {\ vec {f}} \ colon V_ {E} \ to V_ {F}}$ ${\ displaystyle V_ {E}}$ ${\ displaystyle V_ {F}}$

Conversely, every image that can be represented in this way is an isometric drawing. Isometrics of a Euclidean point space are also called movements .

Example of a non-surjective isometry

With regard to the discrete metric , every injective mapping is an isometry. Thus the mapping defined by is a non-surjective isometry. ${\ displaystyle f \ colon \ mathbb {N} \ rightarrow \ mathbb {N}}$ ${\ displaystyle f (n) = 2n}$

Another example of a non-surjective isometry is the inclusion of a real subset of any metric space , where the metric is given by. ${\ displaystyle \ iota \ colon Y \ subset X}$ ${\ displaystyle (X, d)}$ ${\ displaystyle Y}$ ${\ displaystyle d \ mid _ {Y \ times Y}}$

Other properties

From the definition it follows immediately that every isometry is continuous.
Every isometry is even Lipschitz continuous , i.e. in particular uniformly continuous . Isometrics can thus be continuously continued to the conclusion when the image space is complete .
Every metric space is isometrically isomorphic to a closed subset of a normalized vector space, and every complete metric space is isometrically isomorphic to a closed subset of a Banach space .
Every isometry between two Euclidean spaces is also given angles , area and volume .
If two figures can be mapped onto each other in Euclidean spaces using an isometry, then the figures are called isometric. Two figures that can be mapped onto one another through a movement are called congruent .
In general, every isometric drawing between metric spaces is given the Hausdorff dimensions .

literature

Siegfried Bosch : Linear Algebra . 2nd revised edition. Springer, Berlin et al. 2003, ISBN 3-540-00121-2 , ( Springer textbook ).
Fritz Reinhardt, Heinrich Soeder: dtv atlas for mathematics. Boards and texts . tape 1 . 9th edition. Deutscher Taschenbuch Verlag, Munich 1991, ISBN 3-423-03007-0 .

Individual evidence

↑ Jussi Väisälä: A proof of the Mazur-Ulam theorem. Retrieved April 14, 2014 .
^ Stanisław Mazur, Stanisław Ulam: Sur les transformationes isométriques d'espaces vectoriels normés . In: CR Acad. Sci. Paris . tape 194 , 1932, pp. 946-948 .

[1] Jussi Väisälä: A proof of the Mazur-Ulam theorem. Retrieved April 14, 2014 .

[2] Stanisław Mazur, Stanisław Ulam: Sur les transformationes isométriques d'espaces vectoriels normés . In: CR Acad. Sci. Paris . tape 194 , 1932, pp. 946-948 .