# Questions tagged [isometry]

An isometry is a map between metric spaces that preserves the distance. This tag is for questions relating to isometries.

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### Show that $T$ is linear, bounded and Find $||T||$ [closed]

For $T:l² ----> l²$ defined as $T(e1, e2, e3, .....) = (e4, e5, e6, ....)$ Show that $T$ is linear, bounded and find $||T||$ Find $T^*$ 3.Is $T$ an isometry? 4.Is $T$ unitary? I have taken ...
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### Confused by the path I am asked to follow in order to solve the killing equation on S2.

I am asked to find the Killing vector fields on $S^2$ where the line element is given by $ds^2=d\theta\otimes d\theta +\sin^2\theta d\phi\otimes d\phi$. I know how to solve this problem by considering ...
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### Metric Space with No Local Isometries?

We know there are metric spaces with no non-trivial (global) isometries. For example, consider the Euclidean plane $(\mathbb{R}^2, d)$. Let $X \subset \mathbb{R}^2$ be the set $\{(0,0), (1,0),(0,2)\}$....
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### What's the other way to think about SE2 and SE3? My brain only seems to go one way

I work at a robotics company and we have littered throughout our code transforms of the form t_a_b. I continually think of t_a_b ...
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### Isometry Between C[0,1] and C[1,2]

This is not homework, I am just trying to make something clear for myself. I am trying to show that $C([0,1])$ and $C([1,2])$ are isometric. Here both spaces are endowed with the standard sup metric. ...
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1 vote
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### Proving $\phi$ is a smooth map and constructing an explicit isometry

Consider a Lorentzian manifold $(\zeta,g)$ with metric: $$g=\frac{dudv}{uv}-\frac{dr^2}{r^2}-\frac{dw^2}{w^2}.$$ For $u,v,w,r,>0$. Suppose we take a Cauchy foliation of $\zeta,$ called $\mathcal F$,...
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### Does the Mobius Strip have an homogenous embedding?

So in this question I’m trying to do two things at once. 1. Define what a “homogenous embedding” is by describing what it is like and then furthermore ask if such an embedding exists for the mobius ...
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### Orientation reversing diffeomorphism but no isometry?

Is it possible that an oriented Riemannian manifold $(M,g)$ with a large isometry group $\text{Isom}(M,g)$ has an orientation reversing self-diffeomorphism but no orientation reversing self-isometry, ...
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### Local Isometries from the disk to itself are Mobius maps?

Let $f:\mathbb{D} \rightarrow \mathbb{D}$ be a local isometry. I want to show that is an automorphism (i.e. a Mobius map). Here's one brief argument I've come across: "By the Pick Theorem (in ...
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