Questions tagged [isometry]
An isometry is a map between metric spaces that preserves the distance. This tag is for questions relating to isometries.
1,113
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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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2
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51
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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 ...
2
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0
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85
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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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1
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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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prove that every isometry Is also continous [duplicate]
Let $ (X,d_x) $ and $(Y,d_y)$ be metric spaces, an isometry $f:(x,d_x)\rightarrow (y,d_y)$
Is Always continous
Proof:
A function Is an isometry if keeps distances unaltered therfore $ \forall x_1,x_2\...
0
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1
answer
57
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Why Does the Existence of Eigenvalue 1 in Odd Dimensions does Extend to Even Dimensions? ($\mathbb{R}$ vector space)
Let $(V,\langle.,.\rangle)$ be a Euclidean vector space defined over $\mathbb{R}$ of odd dimension $n $ and let $f : V \rightarrow V$ an orthogonal mapping with $\operatorname{det}(f)=1 $. Then the ...
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1
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61
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How do I find the fixed points of one isometry, from the fixed points of another isometry?
I have an isometry group $G$ which acts on units of a ring $R$ (not neessarily the full isometry group). I know the set $P_f$ of fixed points in $R$ of one member $f$ of $G$. Can I use the group ...
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Show that $T$ is a topological embedding of $X$ in itself.
Let $X$ be the set of odd positive fractions with denominator $3$ in lowest terms, except for $\frac13$.
$X=\{x\in\Bbb Q^+:3x\in3\Bbb N\pm1\}\setminus\{\frac13\}$
If $a,b$ are 2-adic units, then any 2-...
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51
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What is the sum of the 2-adic values of the orbits of the affine group over 2-adic integers restricted to $ax+b$ where $2^{\nu_2(x)}=2^{\nu_2(b)}$?
Let $\textrm{aff}(\Bbb Z_2)$ be the affine group over 2-adic integers, defined as the linear polynomials $ax+b$ where $a\in\Bbb Z_2^\times$ and $b\in\Bbb Z_2$.
A very interesting subset of this group ...
- 9,115
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0
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41
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Sets invariant under isometries of $\mathbb{R}^n$
Let $f : \mathbb{R}^n \to \mathbb{R}^{n}$ be an isometry. Is it true that $f(U) \subseteq U$ implies that $U \subseteq f(U)$ for an arbitrary subset $U$ of $\mathbb{R}^n$?
It is intuitively clear to ...
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isomorphism between $l^{2}(Z)$and $L^{2}(S,C)$
i want to prove that $l^{2}(Z)$is isometric to $L^{2}(S,C)$, $S=\{z\in C,|z|=1\}$
may be the answer is to take an operator A such that :$Af=c_{n}$ where $c_{n}$ is Fourier coefficient of f ? is is ...
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1
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1
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Conformal group of the Weeks manifold
The Weeks manifold https://en.wikipedia.org/wiki/Weeks_manifold is an arithmetic hyperbolic 3-manifold known for having the smallest volume of any closed orientable hyperbolic 3-manifold (if you ...
- 7,146
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2
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172
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Necessary condition for the product of two rotations of the plane to be a rotation
I know that any rotation of $\mathbb{R}^2$ can be expressed as the product of two reflections in non-parallel lines, and hence the product of two rotations can be written as $R_{L_4}R_{L_3}R_{L_2}R_{...
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Approximating an isometry with a neural network
I'm trying to link a machine learning framework to more theoretical considerations about linear isometries.
Let's say we have an input dataset $\mathcal{D}=\{(\textbf{x}_1,y_1), (\textbf{x}_2,y_2), \...
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2
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167
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Isometry between vector spaces with different dimensions
Is it possible to have an isometry between two spaces of different dimension? Suppose we have $f : \mathbb{R}^d \rightarrow \mathbb{R}$ with $d \neq 1$ such that $|f(x) - f(y)| = \|x -y\|_2$ for all $...
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57
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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
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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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61
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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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1
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60
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Comparison of terms in local formulation of Christoffel-symbols in relation to isometries.
Let $(M,g),(N,h)$ be semi-riemannian manifolds, and $\varphi \in C^{\infty}(M,N)$ an isometry (hence a diffeomorphism). Let $(U,\psi = (x^1,\ldots,x^n))$ be a coordinate chart of a point $p \in U \...
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Totally geodesic diffeomorphism and isometry.
Suppose $(M,g_M),(N,g_N)$ are two Riemann manifolds of the same dimension, and $f$ is a totally geodesic diffeomorphism between them, is it true that $M,N$ must be isometric (probably not through $f$, ...
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Surfaces in $\mathbb{R}^n$ locally isometric to $S^{n-1}$
As a consequence of the Theorema Egregium we know that $\mathbb{R}^2$ is not even locally isometric to $S^2$. It's then easy to generalise this so that $2$ is any $n\geq 2$. Now, it's easy to come up ...
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How to proof this linear congruent transformation?
First of all I would like to thank you for taking a closer look at the following problem, which unfortunately I have been struggling to solve for days:
Let $W$ be a n-dimensional Euclidean vector ...
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1
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Construct on $\mathbb{R}^2$ a norm $\lVert \cdot \rVert$ such that the metric space ($\mathbb{R}^2 , \lVert \cdot \rVert$) is neither isometric to the
Construct on $\mathbb{R}^2$ a norm $\lVert \cdot \rVert$ such that the metric
space ($\mathbb{R}^2 , \lVert \cdot \rVert$) is neither isometric to the space ($\mathbb{R}^2 , \lVert \cdot \rVert_2$), ...
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If $T\in B(H)$ and let $U,V\in B(H)$ be two isomtries such that $TT^*=UU^*$ and $T^*T=VV^*$. Is it true that $T=USV^*$ for some unitary S?
Let $T$ be a bounded operator on a Hilbert space $H$ and $U,V$ be two isometries such that $T=USV^\ast$ for some unitary $S$. Then $TT^\ast=USV^\ast VS^\ast U^\ast=UU^\ast$ and $T^\ast T=VS^\ast U^\...
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1
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Local isometry between half a disk and the cone of revolution $3(x^2+y^2)=z^2$
This is an exercise from my Differential Geometry course:
Define the function $\Phi:\left]0,2\right[\times \left]-\pi/2,\pi/2\right[\longrightarrow \mathbb{R}^2$, $\Phi(\rho,\theta)=(\rho\cos \theta,\...
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The classification of isometries of hyperbolic space via linear algebra
I am reading Thurston's Three-dimensional Geometry and Topology and in particular the sections on isometries of hyperbolic space. I have read (and mostly understood) the qualitative/geometric proofs ...
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Equivalence of different metrics on orbit space
Let $(M,\mu)$ be a Riemannian manifold with an isometric action of a compact Lie group $G$. If the action is not free, the orbit space $M/G$ is a stratified space
\begin{equation}
M/G = \bigcup_{H<...
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Invariant/basic forms with respect to the closure of a subgroup (of isometries)
Let $(M,g)$ be a Riemannian manifold. By a theorem of Myers and Steenrod, the group of isometries $Iso(M)$ is a Lie group. By the closed subgroup theorem, the closure of any subgroup of isometries is ...
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1
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1
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If $f_n$, f are isometries, then $\underset{n\to\infty}{\lim}\overset{\infty}{\underset{i=0}{\sum}}2^{-i-1}d(f_n(x_i), f(x_i))=0$.
The Problem: Suppose $(X, d)$ is a complete separable metric space, and $\{x_i\}\subseteq X$ is a countable dense subset. Suppose $(f_n)$ is a sequence of isometries of $X$ such that $\underset{n\to\...
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1
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The image of a $C^*$-algebra is closed under an isometry
I need to show that $\mathcal{A}$ closed and $\|{\phi(A)}\|=\|{A}\|$ imply that $\phi(\mathcal{A})$ is closed in $C(\Delta(X))$ (I don't think this is necessary but $\mathcal{A}$ is a $C^*$-algebra, $\...
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Proving that $S$ is localy isometric to a surface of revolution given the first fundamental form
I'm having some trouble with the following exercise:
Let $S$ be a surface that admits a global parametrization $\phi(u,v)$ such that the coefficients of the first fundamental form are
$$E:=\left<\...
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View the unitary group $U(n)$ as the group of linear isometries of $\mathbb{C}^n$.
The Statement: Viewing $\mathbb{C}^n$ as an $n$-dimensional Hilbert space, we can view $U(n)$ as the group of linear isometries of $\mathbb{C}^n$.
(Definition: The unitary group $U(n)$ consists of all ...
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Uncertain about the statement "equivalent condition for an isometry of Riemannian manifolds" in Lee's Intro to Riemannian Manifolds
In Professor Lee's Introduction to Riemannian Manifolds, second edition on page 12,
the first paragraph on Isometries reads
Suppose $(M,g)$ and $(\tilde{M},\tilde{g})$ are Riemannian manifolds with ...
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3
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A question about trace class operatos and partial isomertry
I'm a beginner in operator theory and I'm trying to understand just one step in a proof,which claims that if $H$ is a Hilbert space, $A\in B_1(H)$(that is , $A$ is a trace-class operator), and $U$ is ...
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Classification of isometry groups
Let $M$ be a Riemannian manifold. Are there results on the classification of possible isometry groups of $M$? Does the classification of three-manifolds lead to such a classification of isometry ...
4
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1
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Elliptic Isometries have bounded orbits
I was reading Bridson and Haefligers book Metric Spaces of Non-Positive Curvature and was having trouble understand a proof of a proposition.
In Chapter II, Proposition 6.7, it is stated that
Let $X$ ...
3
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2
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92
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Show that there exists a $\alpha \in \mathbb{C}$ for which $\alpha f: V \rightarrow V$ is an isometry.
Task:
Let $(V,\langle.,.\rangle)$ be a unitary vector space, $f \in \operatorname{Hom}_{\mathbb{C}}(V, V)=\operatorname{End}_{\mathbb{C}}(V) $ an endomorphism $ \neq 0 $ with the property that for all ...
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$(\ell^{1})^{*}$ isometrically isomorphic to $\ell^{\infty}$ as a corollary of Riesz representation Theorem.
I'm following "A Course in Functional Analysis, Conway" and as a corollary of the Riesz theorem (Example 5.9) he states what I have written in the title.
If I consider $\mathbb{N}$ with the ...
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1
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83
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endomorphism and self adjoint operators and isometry
Let $F=\mathbb{R}$ or $F=\mathbb{C}$, let $(V,\langle.,.\rangle)$ be a finitely generated $K$-vector space with scalar product, let $f \in \operatorname{End}_{F}(V)$. Show that:
a) If $f$ is self-...
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0
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54
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Is there an isometric embedding from the Euclidean plane to the sequence space with the $\ell_1$ metric?
I am considering two metric spaces:
The Euclidean plane $\mathbb R^2$, equipped with the Euclidean distance metric $d((x, y), (x^\prime, y^\prime)) = \sqrt{(x - x^\prime)^2 + (y - y^\prime)^2}$, and
...
- 115
1
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0
answers
59
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Locally isometric surfaces
I have to decide whether the surface $S$ parametrized by
\begin{align*}\mathbf{x}:\mathbb{R}\times \left]-\pi,\pi\right[&\longrightarrow S\\
(u,v)&\longmapsto ((2+\sin u)\cos v,(2+\sin u)\sin ...
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0
votes
1
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40
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Equivalent metrics and equicontinuity [closed]
Let $(X,d)$ be a compact metric space, and $\varphi$ be an equicontinuous dynamical system ($\forall\, \varepsilon>0$, there exists $\delta>0$ such that $d(x,y)<\delta\Rightarrow d(\varphi^nx,...
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1
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1
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118
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Orbifold signature and fundamental tile of Escher's Circle Limit IV
In Conway, Burgel and Goodman-Strauss' book The symmetries of things, Chapter 17, the following
picture by Escher
was analysed using orbifold notation. It's a hyperbolic pattern in the Poincare disk.
...
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2
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1
answer
95
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When do two translations in the hyperbolic plane commute?
If I'm correct, to define a hyperbolic translation you need an ideal point $p$ as a source, a different ideal point $q$ as a sink and a length $d \ne 0$ along the geodesic that joins $p$ and $q$. Let'...
- 343
1
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1
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55
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Isometries between two non parallel straight lines.
for the past few days I've been studying Euclidean geometry and I was working on some problems concerning Euclidean transformations which are just isometries. The problem is the following: Let $l$ and ...
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2
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1
answer
122
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4th moment of a Wiener stochastic integral with Ito isometry property?
Suppose we have a Wiener stochastic integral
$$\int_{t-h}^t f(r_v) d W_v, \tag{1}$$
It is well known that, by the Ito isometry property,
$$\mathbb E \left [\left(\int_{t-h}^t f(r_v) d W_v\right)^2 \...
- 21
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votes
1
answer
42
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Isometry Invariance of the Inner Product in $L^2(M)$
Let $(M,g)$ is oriented Riemmanian manifold and endowed with a Riemannian metric $g$. Let $dV$ is the Riemannian volume form
Let us consider the function space $L^2(M)$. We define the inner product as ...
- 6,483
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0
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43
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Isometries of S2 in spherical coordinates
Can anyone give equations of isometries of $S2$ in spherical coordinates $(\theta, \varphi)$? In other words, we need to find isometries of $S2$ with respect to metric:
$g=d\theta \otimes d\theta+ \...
