#REDIRECT [[Quasi-isometry]]
28 bytes (2 words) - 10:09, 17 December 2017
...{A}\right) = B$, that is, if $f$ is a bijection, then $f$ is said to be an isometry from $A$ onto $B$, and $A$ and $B$ are said to be in isometric corresponden
An isometry of real Banach spaces is an affine mapping. Such a linear isometry is realized by (and called) an [[Isometric operator|isometric operator]].
2 KB (277 words) - 06:49, 14 January 2017
If the isometry of surfaces implies congruence, more exactly, if any surface $ F $
...ty that the isometry is necessarily represented by a restriction of a self-isometry of the ambient space, then $ F _ {0} $
2 KB (264 words) - 22:13, 5 June 2020
corresponding under the isometry is less than $ \pi $,
...This fact has enabled one to reduce in a number of cases the study of the isometry of $ F _ {1} $
3 KB (390 words) - 17:45, 4 June 2020
See also [[Quasi-isometry|Quasi-isometry]].
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quasi-isometry.
...sometry. Some authors (see, e.g., [[#References|[a3]]]) use the word quasi-isometry to denote a mapping having the property above, with the further condition t
3 KB (411 words) - 08:09, 6 June 2020
which is a [[Quasi-isometry|quasi-isometry]] invariant of the [[Metric space|metric space]] $ ( G,d _ {A} ) $,
3 KB (362 words) - 09:44, 14 April 2024
...orms (in another terminology, isometry) and, correspondingly, isomorphism (isometry) of Hermitian spaces (in particular, automorphism). All automorphisms of a
...d isometry of two non-degenerate Hermitian forms over $R$ is equivalent to isometry of the quadratic forms over $R_0$ generated by them; this reduces the class
5 KB (831 words) - 17:13, 9 October 2016
...of a single preserved distance for some mapping $f$ implies that $f$ is an isometry (cf. [[#References|[a1]]]).
...} ^ { 2 }$ to $\mathbf{R} ^ { 3 }$ preserving unit distance necessarily an isometry (cf. [[#References|[a17]]])?
8 KB (1,153 words) - 16:46, 1 July 2020
...nometry is called a dimetry, and if three distortion indices are equal, an isometry; if all distortion indices are different, it is called a trimetry. The prin
1 KB (194 words) - 16:56, 7 February 2011
is an isometry (cf. [[Isometric operator|Isometric operator]]) on a Hilbert space $ {\ma
an isometry on $ {\mathcal H} $
6 KB (815 words) - 09:51, 26 March 2023
...s with respect to completion, which is a more general concept than that of isometry with respect dissection.
4 KB (583 words) - 15:04, 9 April 2014
corresponding in isometry, the trihedra formed by the tangent vectors $ x _ {u} , x _ {v} $
...efore it plays the same part as the rotation indicatrix when examining the isometry of convex surfaces.
4 KB (541 words) - 15:08, 19 January 2021
Any isometry between two subspaces $ F _ {1} $
A second consequence of Witt's theorem may be stated as follows: The isometry classes of non-degenerate symmetric bilinear forms of finite rank over $
4 KB (558 words) - 16:18, 6 June 2020
...the rotation diagrams for these surfaces are the same, and the base of the isometry for the new surfaces has the same spherical image as the original ones. For
In particular, if the base of the isometry for $ S $
4 KB (660 words) - 19:40, 3 January 2021
is a [[Quasi-isometry|quasi-isometry]].
4 KB (659 words) - 08:29, 6 June 2020
.... D. Sullivan introduced the concept of quasi-self-similarity. A $K$-quasi-isometry is defined by a function $f$ acting on a metric space $M$ with metric $d$ s
...ch that multiplication by $1/r$ of $F\cap D_r(x)$ maps into $F$ by a quasi-isometry for all $r<r_0$ and all $x\in F$. (Here $D_r(x)$ is the open ball centred a
4 KB (542 words) - 19:03, 16 April 2014
...{C_{0}}(Y) $. The Banach–Stone theorem asserts that any linear surjective isometry $ T: C(X) \to C(Y) $ is of the form above. Here, if $ X $ is not necessaril
5 KB (819 words) - 17:08, 6 January 2017
and an isometry $ T: V \rightarrow V $.
is the characteristic polynomial of the isometry $ T $.
11 KB (1,631 words) - 17:45, 4 June 2020
...d is locally isometric to the [[Helicoid|helicoid]] and in fact such local isometry can be achieved as endpoint of a continuous one-parameter family of isometr
2 KB (303 words) - 20:07, 12 November 2023
...dentified boundaries are isometric and identification is carried out by an isometry (cf. [[Isometric mapping|Isometric mapping]]). The distance between two poi
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...the helicoid is isometric to the [[Catenoid|catenoid]]. In fact such local isometry can be achieved as endpoint of a continuous one-parameter family of isometr
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is a linear [[Isometric mapping|isometric mapping]] (linear isometry) of $ A $
is an algebra isometry of $ A $
8 KB (1,167 words) - 11:37, 12 January 2021
...h point $p\in M$ is an isolated fixed point of some holomorphic involutory isometry $s_p$ of $M$. The component of the identity of the group $G$ of holomorphic
2 KB (388 words) - 23:00, 12 December 2020
This form is independent (apart from an isometry) of the choice of the Witt decomposition of $ V $.
anisotropic. Moreover, the isometry classes of $ ( V _ {t} , q _ {t} ) $,
8 KB (1,152 words) - 08:29, 6 June 2020
of which is an isolated fixed point of some involutory isometry $ S _ {p} $
...simply-connected irreducible globally symmetric Riemannian spaces up to an isometry is equivalent to the classification of irreducible orthogonal symmetric Lie
9 KB (1,321 words) - 19:42, 5 June 2020
this fact is a generalization of the isometry of a horosphere in Lobachevskii space to a Euclidean space.
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Up to an isometry there exists a unique complete simply-connected $n$-dimensional Riemannian
4 KB (626 words) - 05:24, 29 May 2016
...c domain $D$ in a complex Banach space there exists a unique (up to linear isometry) $\operatorname{JB} ^ { * }$-triple $E$ whose open unit ball is biholomorph
A bijective linear operator between $\operatorname{JB} ^ { * }$-triples is an isometry if and only if it respects the Jordan triple product.
10 KB (1,477 words) - 06:10, 26 March 2024
...ncyclopediaofmath.org/legacyimages/b/b110/b110090/b11009041.png" /> is the
isometry group of <img align="absmiddle" border="0" src="https://www.encyclopediaofm
10 KB (1,395 words) - 06:44, 9 October 2016
along curves on the base is an isometry of the fibres as unitary spaces. For every Hermitian metric there is a conn
4 KB (681 words) - 03:41, 21 March 2022
...y, also (b) implies (a), which involves a Borel isomorphism (rather than isometry or homeomorphism) between two metric spaces.
5 KB (655 words) - 20:33, 18 February 2012
...eodesic and the mapping $\psi : ( u , v ) \rightarrow ( 2 u , 2 v )$ is an isometry (cf. also [[Isometric mapping|Isometric mapping]]). Defining $x \sim y$ as
4 KB (606 words) - 19:47, 24 November 2023
...s the maximum of the $\dim(G(M, ds^2))$. For a description of the possible isometry groups of dimensions $\ge 2+(n-1)(n-2)/2$ cf. [[#References|[a2]]], [[#Refe
19 KB (2,034 words) - 04:04, 15 February 2024
...s spaces uniquely up to a piecewise-linear homeomorphism and even up to an isometry; on the other hand, by the topological invariance of the torsion, they also
4 KB (668 words) - 18:39, 13 January 2024
...the plane (axis) of the symmetry (see ). Any orthogonal transformation, or isometry, is the composite of a finite number of reflections.
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...encyclopediaofmath.org/legacyimages/r/r082/r082150/r08215057.png" /> is an
isometry, then for any point <img align="absmiddle" border="0" src="https://www.ency
....org/legacyimages/r/r082/r082150/r082150105.png" />; it turns out to be an
isometry and is called the Levi-Civita parallel displacement. The result of the tran
55 KB (7,564 words) - 16:56, 7 February 2011
where $\phi$ is an isometry with $\phi ^ { 2 } = \operatorname{id}$. Note that in dimension $m = 4$, th
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is a partial isometry (i.e. $ u ^ {*} u $
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is an isometry in the $ Q $-
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for all $v\in V$. This isomorphism is called an isometry of the form and, if $V=W$ and $f=g$, a metric automorphism of the module $V
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in the isometry group of $ M $(
7 KB (1,081 words) - 16:58, 1 July 2020
by a motion (that is, an isometry, cf. [[Isometric mapping|Isometric mapping]]) of the plane. If $ A x ^ {
8 KB (1,242 words) - 07:04, 6 May 2022
...do-Riemannian (affine) space if one can assign to every point $x \in M$ an isometry (affine transformation) $S_x$ of $M$ such that $S_x^2 = id$ and $x$ is an i
8 KB (1,254 words) - 14:07, 2 January 2014
Usually, an isometry of a Riemannian space is used as a synonym for motion, while the isometries
8 KB (1,131 words) - 07:48, 23 May 2022
...>[a10]</TD> <TD valign="top"> C.B. Yue, "The ergodic theory of discrete isometry groups on manifolds of variable negative curvature" ''Trans. Amer. Math. S
8 KB (1,145 words) - 22:11, 5 June 2020
s implies isometry (Mostow's rigidity theorem).
is an isometry, then the function $ \delta _ \gamma ( x) = \rho ( x, \gamma x) $
25 KB (3,732 words) - 22:15, 7 June 2020
...{ - 1 }$ is continuous (by the Banach theorem). Consequently, $L$ is a co-isometry, that is, $L ^ { * } = L ^ { - 1 }$, where $L ^ { * }$ is the [[Adjoint ope
14 KB (2,151 words) - 16:39, 2 February 2024
...mapsto \int _ { \Omega } x x ^ { \prime } d \mu$ on $X$ is an order-linear isometry from the Köthe dual space $X ^ { \prime }$ onto $X ^ { * }_{c}$. In this w
16 KB (2,441 words) - 20:16, 25 January 2024
...e called spherical subspaces of $S$. A mapping $f : S \rightarrow S$ is an isometry of $S$ if $d ( x , y ) = d ( f ( x ) , f ( y ) )$ for all $x$, $y$. Motions
12 KB (1,960 words) - 17:44, 1 July 2020
...\tilde\rho)$ is a complete metric space and $i:X\rightarrow\tilde X$ is an isometry of $(X,\rho)$ onto an everywhere-dense subset in $(\hat X,\tilde\rho)$ ($(\
...(y))\leq d(x,y)$ for all $x,y\in I$). $I$ is characterized, up to a unique isometry, as an essential extension of $X$ which has no further essential extension.
33 KB (5,289 words) - 19:37, 25 March 2023
...tural inner products on $R$ and $\Lambda$ that make $\operatorname{ch}$ an isometry.
14 KB (2,001 words) - 10:09, 11 November 2023
...tral subspaces $M ^ { U } ( E )$ generalize to arbitrary Banach spaces and isometry groups, satisfying the basic assumptions above, giving the familiar spectra
14 KB (2,151 words) - 17:43, 1 July 2020
for all $a \in A$. If the involution in $A$ is an isometry, i.e. if $\norm{a^*}=\norm{a}$ for all $a \in A$, then
14 KB (2,346 words) - 22:48, 29 November 2014
...does $\operatorname { spec } ( M , \Delta )$ determine $( M , g )$, up to isometry (see also below), has been studied quite extensively. In particular, the an
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Non-compact Euclidean space forms have been classified (up to an isometry) only in dimensions 2 and 3. In particular, a two-dimensional non-compact E
18 KB (2,658 words) - 19:36, 18 January 2024
...E _ { i } ^ { * } \xi = \xi ^ { \prime }$. Indeed, each $E_i$ is a partial isometry, satisfying $E _ { i } ^ { * } E _ { j } + E _ { j } E _ { i } ^ { * } = \d
19 KB (2,677 words) - 17:44, 1 July 2020
...d></tr><tr><td valign="top">[a23]</td> <td valign="top"> S. Tanno, "On the isometry of Sasakian manifolds" ''J. Math. Soc. Japan'' , '''22''' (1970) pp. 579–
17 KB (2,475 words) - 12:23, 12 December 2020
...S\to T_{a_1}S$ which sends $v_0$ to $v_1$ and is an orientation-preserving isometry, is uniquely defined by these two conditions. This defines the "parallel tr
16 KB (2,769 words) - 08:44, 12 December 2013
1) An important classification tool for metric spaces is the concept of quasi-isometry: Two metric spaces are quasi-isometric if there are bijective mappings betw
18 KB (2,803 words) - 16:46, 1 July 2020
...es in which the geodesic symmetry with respect to an arbitrary point is an isometry. These spaces always have a transitive group of isometries and may be class
30 KB (4,323 words) - 19:35, 5 June 2020
...ransport is usually assumed to be compatible with this structure, i.e., an isometry.</ref> ''[[parallel transport]] map'' $T_\gamma:F_{b_0}\to F_{b_1}$ describ
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...or fields, with two such fields being equivalent if there exists a local [[isometry]] conjugating these fields (or, what is equivalent in this case, their pote
37 KB (5,881 words) - 19:10, 24 November 2023
is an isometry
64 KB (9,418 words) - 12:44, 8 February 2020