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
6 KB (1,157 words) - 08:58, 9 December 2016
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
14 KB (2,008 words) - 16:55, 1 July 2020
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