Linked Questions

12
votes
1answer
11k views

A isometric map in metric space is surjective? [duplicate]

Possible Duplicate: Isometries of $\mathbb{R}^n$ Let $X$ be a compact metric space and $f$ be an isometric map from $X$ to $X$. Prove $f$ is a surjective map.
7
votes
2answers
357 views

Let $f: X \to X$ be such that $d(f(x), f(y)) = d(x, y)$ for all $x, y \in X$. To show that $f$ is onto. [duplicate]

Let $(X, d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x), f(y)) = d(x, y)$ for all $x, y \in X$. To show that $f$ is onto. Since the function $f$ satisfies $d(f(x), f(y)) = d(x, ...
1
vote
1answer
2k views

isometry $f:X\to X$ is onto if $X$ is compact [duplicate]

Possible Duplicate: Isometries of $\mathbb{R}^n$ A question about isometry A isometric map in metric space is surjective? How to prove that isometry $f:X\to X$ is onto if $X$ is compact?? ...
2
votes
0answers
62 views

Compact metric space [duplicate]

Let $(X,d)$ be a compact metric space and $f : X \to X$ be isometric, i.e. for every $x,y \in X$ : $ d(f(x),f(y)) = d(x,y) $. How I can show the following? $f(X) = X$ My first thought was to show the ...
6
votes
4answers
1k views

Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective).

Let $(X,d)$ be a compact metric space. Let $f: X \to X$ be such that $d(f(x),f(y)) = d(x,y)$ for all $x,y \in X$. Show that $f $ is onto (surjective). If $f$ is not onto then there exist a $p \in X$ ...
4
votes
4answers
2k views

Prove that every isometry on $\mathbb{R}^2$ is bijective

Let $d(x,y)=\sqrt{(x_1-y_1)^2+(x_2-y_2)^2}$ for $x=(x_1,x_2), y=(y_1,y_2)$. A isometry of $\mathbb{R^2}$ is an image $f:\mathbb{R^2}\to\mathbb{R^2}:d(x,y)=d(f(x),f(y))$. Show that every isometry is ...
0
votes
1answer
2k views

Prove that an isometry is a bijection

attempt of solution: In order to show a bijection, we require two steps: 1) show that it is injective 2) show that it is surjective I have already proved the part to show injectivity but i am not ...
3
votes
0answers
524 views

Distance preserving map on finite dimensional normed space E is a bijection

The problem is to show that for $E$ a finite dimensional normed space and $T:E\to E$ a map (not necessarilly linear) such that $\Vert T(x)-T(y)\Vert =\Vert x-y\Vert \forall x,y\in E$is a bijection. 1-...
1
vote
1answer
132 views

Show that $f$ is onto [duplicate]

Let $(X,d)$ be a compact metric space. Let $f:X\rightarrow X$ be such that $d(f(x),f(y))=d(x,y)$ for all $x,y\in X$. Show that f is onto. Hint: Fix $y\in X $and $ x_1\in X$, define $x_n=f(x_{n-1})$, ...
3
votes
1answer
118 views

Ratio distance similarity transformations: $|\varphi (a)- \varphi (b)| = k|a-b|$

I am struggling with the following group geometry question. I am given that: A simililarity transformation is a non-constant map $\varphi : \mathbb{R^2} \to \mathbb{R^2}$ that leaves the ratios of ...
0
votes
0answers
107 views

A function $\mathbb{R}^n\to\mathbb{R}^n$ that preserves distances must be a linear map followed by a translation [duplicate]

Possible Duplicate: Are isometric normed linear spaces isomorphic? $ f: \mathbb{R}^n \to \mathbb{R}^m $ preserving distances Consider the set of all functions $\varphi : \mathbb{R}^n \...
1
vote
2answers
60 views

Isometry in a compact space is surjective

Let's say we have a compact metric space $(X,d)$ and a function $f: X\to X$ satisfying $d(f(x),f(y)) = d(x,y)$, $x,y\in X$. How can we show that this function is sujective in a simple way.
1
vote
1answer
92 views

Classification of isometries of $\mathbb{R}^n$

The problem is trivial yet I got stuck and have a problem with justification of the fact $\text{Isom}(\mathbb{R}^n,g=\sum_{i=1}^n (dx_i)^2)=\mathbb{R}^n \rtimes O(n)$. Suppose $f:\mathbb{R}^n \...
2
votes
1answer
81 views

If Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix then $f(x) = Ox +b $ [closed]

Let $f : \Bbb R^n \to \Bbb R^n$ be a $C^1$ map such that it's Derivative at each point $x \in \Bbb R^n$ is an orthogonal matrix i.e. $Df_x \in O(n,\Bbb R) \text{ , } \forall x \in \Bbb R^n$ . Then ...
1
vote
2answers
57 views

Examples of topology preserving transforms

I was trying to think of a transformation in Euclidean space which is topology preserving but not affine. I fact I could not think of even a non affine isometric transform in Euclidean space. Here ...

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