# What is Isometry: Definition and 47 Discussions

In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.

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1. ### I A = norm-preserving linear map (+other conditions) => A = lin isometry

I'm studying "Semi-Riemannian Geometry: The Mathematical Langauge of General Relativity" by Stephen Newman. Theorem 4.4.4 in that book: The proof of part 2 is given like this: Seems a bit incomplete. I'd like to know if my approach is correct: \langle A(v+tw),A(v+tw)\rangle=\langle...
2. ### I Minkowski Spacetime KVF Symmetries

Hi, reading Carrol chapter 5 (More Geometry), he claims that a maximal symmetric space such as Minkowski spacetime has got ##4(4+1)/2 = 10## indipendent Killing Vector Fields (KVFs). Indeed we can just count the isometries of such spacetime in terms of translations (4) and rotations (6). By...
3. ### I Show isometry and find geometric meaning

The matrix ##A## in question is ##\dfrac{1}{3} \left(\begin{array}{rrr} -2 & 1 & -2 \\ -2 & -2 & 1 \\ 1 & -2 & -2 \end{array}\right)## One can easily verify that ##AA^t=I##, hence an isometry. To find its geometric meaning, one can proceed to find ##U=\text{ker} \ (F-I)=\text{ker} \...
4. ### A AdS##_3## Cylinder: Killing Vectors & Isometry Group

The isometry group of the anti-de Sitter spacetime is ##SO(d-1,2)##, which has a total of ##\frac{1}{2}d(d+1)## isometries. For the three-dimensional anti-de Sitter spacetime, these are ##6## isometries. These isometries have corresponding Killing vectors, which in global coordinates, are given...
5. ### B (Proof) Two right triangles are congruent.

Hi, the question is from Serge Lang - Basic mathematics, Page 171 exercise 6. Thing to prove: Let ΔPQM and ΔP'Q'M' be right triangles whose right angles are at Q and Q', respectively. Assume that the corresponding legs have the same length: d(P,Q)=d(P',Q') d(Q,M)=d(Q',M') Then the right...
6. ### I Dimension of the group O(n,R) - How to calc?

Hi, I want to find the number of parameters needed to define an orthogonal transformation in Rn. As I suppose, this equals the dimension of the orthogonal group O(n,R) - but, correct me if I'm wrong. I haven't been able to figure out how to do this yet. If it helps, I know that an orthogonal...
7. ### I Why is this Isometry a rotation?

Hello, i need a little help. Did someone have an idea how to prove this? Thanks in advance. Be ##\Phi## an direct isometry of the euclidean Space ##\mathbb{R}^3## with ##\Phi (\begin{pmatrix} 2\\0 \\1 \end{pmatrix})##=##\begin{pmatrix} 2\\1 \\0 \end{pmatrix}## and ##\Phi (\begin{pmatrix}...
8. ### A Q: Scalar Boundary Condition & U(1) Isometry - Lewkowycz & Maldacena

I have a simple question about Lewkowycz and Maldacena's paper http://arxiv.org/abs/1304.4926v2'][/PLAIN] [Broken] http://arxiv.org/abs/1304.4926v2 In section 2, they consider the scalar field in BTZ background ground and require boundary condition of the scalar field, $\phi \sim e^{i\tau}$ ...
9. ### Isometry and isomorphism

I have read a definition of isomorphism as bijective isometry. I was also showed a definition that isomorphism is a bijective map where both the map and its inverse are bounded (perhaps only for normed spaces??). This does not seem to be the same thing as an isometry. For example, the poisson...
10. ### Diffeomorphism and isometry

Let f:p\mapsto f(p) be a diffeomorphism on a m dimensional manifold (M,g). In general this map doesn't preserve the length of a vector unless f is the isometry. g_p(V,V)\ne g_{f(p)}(f_\ast V,f_\ast V). Here, f_\ast:T_pM\to T_{f(p)}M is the induced map. In spite of this fact why...
11. ### Isometry group of positive definite matrices

Hi, Consider P the space of n by n positive definite matrices. Let <X,Y>_p = trace(p^-1 X p^-1 Y) where p in P be the metric tensor on P so that it is a Riemannian manifold. The general linear group G acts on P by phi: G X P -> P, phi(g,p) = phi_g(p)=gpg' (g' means g transpose). This a...
12. ### Doubt about rotation isometry on the complex plane

Dear All, In the 2nd paragraph of the attachment can you please explain to me why we are trying to make ## r(z) = z ## and what does "As r is not the identity..." mean?? and how did the line ## L = 0.5*b + ρe^{θ/2} ## come about? Danke...
13. ### Derivation of rotation isometry on the complex plane

Dear all, can you please verify if my derivation of the algebraic formula for the rotation isometry is correct. The handwritten file is attached. The derivation from the book (Alan F beardon, Algebra and Geometry) which is succinct but rather unclear is given below. Assume that f (z) = az + b...
14. ### Some questions about isometry

Isometry is the symmetry s.t. g^\prime_{\mu\nu}(x)=g_{\mu\nu}(x) under the transformation x^\mu\to x^{\prime\mu}(x). This means under infinitesimal transformation x^\mu\to x^\mu+\epsilon \xi^\mu(x) where \epsilon is any infinitesimal constant, the vector field \xi^\mu(x) satisfies Killing...
15. ### Trivial Isometry Group for the Reals

In the following stackexchange thread, the answerer says that there is a Riemannian metric on \mathbb{R} such that the isometry group is trivial. http://math.stackexchange.com/questions/492892/isometry-group-of-a-manifold This does not seem correct to me, and I cannot follow what he is...
16. ### What kind of isometry? A metric tensor "respects" the foliation?

Suppose we have a foliation of leaves (hypersurfaces) with codimension one of some Riemannian manifold ##M## with metric ##g##. For any point ##p## in ##M## we can then find some flat coordinate chart ##(U,\phi) = (U, (x^\mu, y))## such that setting ##y## to a constant locally labels each leaf...
17. ### Are any two geodesics the same after applying an isometry?

Hello, I was wondering the following. Suppose you start with a Riemannian manifold M. Say you know one geodesic. Pushing this geodesic forward through an isometry M -> M gives again a geodesic. Can this procedure give you all geodesics? Thinking of the plane or the sphere it seems...
18. ### MHB Orientation-preserving isometry of R^n

Orientation-preserving isometry I am preparing for an exam, and would like to have a rigorous definition of the following: **Orientation-preserving isometry of $R^n$** I know that it is something like the following (feel free to correct my wording): When the homomorphism \$\pi:M_n...
19. ### Definition of isometry in components form

From the book of Nakahara "Geometry, Topology and Physics": A diffeomorfism ##f:\mathcal{M}\to \mathcal{M}## is an isometry if it preserves the metric: ## f^{*}g_{f(p)}=g_{p} ## In components this condition becomes: ## \frac{\partial y^{\alpha}}{\partial x^{\mu}}\frac{\partial...
20. ### MHB Wizard1's question at Yahoo Answers (Isometry)

Here is the question: Here is a link to the question: Verify that T is an isometry of H...? - Yahoo! Answers I have posted a link there to this topic so the OP can find my response.
21. ### Fourier Series Isometry

Hey guys. I just started a class on Fourier Analysis and I'm having a difficult time understanding this question. Any help would be much appreciated! Homework Statement Verify that the Fourier Isometry holds on [−π, π] for f(t) = t. To do this, a) calculate the coeﬃcients of the orthogonal...
22. ### Hyperbolic Paraboloid and Isometry

If the hyperbolic paraboloid z=(x/a)^2 - (y/b)^2 is rotated by an angle of π/4 in the +z direction (according to the right hand rule), the result is the surface z=(1/2)(x^2 + y^2) ((1/a^2)-((1/b^2)) + xy((1/a^2)-((1/b^2)) and if a= b then this simplifies to z=2/(a^2) (xy) suppose...
23. ### Existence of some special kind of isometry in R^n-1

Homework Statement Is there a theorem that states the following? Let P= \{ P_{1}, . . . , P_{n} \} be the set of n distinct points in \mathbb{R}^{n-1} and P'= \{ P'_{1}, . . . , P'_{n} \} also a set of points in \mathbb{R}^{n-1}. If for all i,j |P_{i} - P_{j}|=|P'_{i} - P'_{j}| then there...
24. ### MHB Surjectivity of an Isometry given the metric space is complete.

Hello, the following is a post that was in progress and I am continuing it here after I received a message saying that most of the members had moved from mathhelpforum here. Me: I have a problem where I am asked to show that for a complete metric space X, the the natural Isometry F:X --> X* is...
25. ### Isomorphism & Isometry: Hilbert Spaces

Hi, I am wondering if all isomorphisms between hilbert spaces are also isometries, that is, norm preserving. In another sense, since all same dimensional hilbert spaces are isomorphic, are they all related by isometries also? Thank you,
26. ### Finding Isometry between Two Parametric Lines in R^3

Homework Statement i was given two parametric equations of two lines in R^3 and asks me to find the isometry between one line and the other knowing that point (a,b,c) of first line is mapped in point (a',b',c') of second line. Homework Equations What i have to find is a 3x3 matrix which...
27. ### Need help for Ito Isometry proof

Hi, There is one step in Ito Isometry proof I don't understand. Bt(w) represents the position of w at time t, then E[(Btj+1(w)-Btj(w))2] = tj+1 - tj. Why is the expectation of the square of the difference of 2 positions equal to their time difference? Any hint, please. Thank you.
28. ### Isometry between euclidian and non-euclidian space

Hi For the moment, I am working on dimensionality reduction of data and the following question has risen during my work. I have a set of data in a non-euclidian space, lying on a manifold for which I know the metric measure d between points with coordinates (mu_1,sigma_1) and...
29. ### Isometry Problem: Does Convergence in Norm Imply Convergence Under Map?

Hello all, My question is if f : X \mapsto Y is an isometry which preserves norm, i.e. \left\| f(x) \right\| _{Y} = \left\| x \right\|_{X} , does this imply \left\| f(x_2) - f(x_1) \right\| _{Y} = \left\| x_2 - x_1 \right\|_{X} ? Or, essentially is it sufficient to gurantee...
30. ### Extend Isometry from Semi-normed to Normed Space

Hello all, May someone help me on this question: Suppose the map F is an isometry which maps a dense set H of a semi-normed space \mathcal{H} to a normed space \mathcal{G} , now the theorem said we can extend this isometry in a unique manner to a linear isometry of the semi-normed...
31. ### Geometry/Topology Isometry question

Homework Statement Let P, Q, P' and Q' be 4 points in the euclidean plane where distance(P,Q) = distance(P', Q') and the distance is not 0. Show there are precisely 2 isometries \varphi so \varphi(Q) = Q^{l} and \varphi(P) = P^{l} Homework Equations -The Attempt at a Solution So I know (I...
32. ### Prove Dual Space Isometry: X = C[0,1] Sup Norm

Homework Statement For the Banach space X = C[0,1] with the supremum norm, fix an element g \in X and define a map \varphi_g : X \to \mathbb{C} by \begin{align*} \varphi_g(h) := \int^1_0 g(t) h(t) dt, \qquad h \in X \end{align*} Define W := \{ \varphi_g | g \in X \}. Prove...
33. ### Isometry group for L1 norm

Hi, I've been trying to show that the set of matrices that preserve L1 norm (sum of absolute values of each coordinate) are the complex permutation matrices. Complex permutation matrix is defined as permutation of the columns of complex diagonal matrix with magnitude of each diagonal element...
34. ### Why is an isometry bijective?

A function from the plane to itself which preserves the distance between any two points is called an isometry. Prove that an isometry must be a bijection. To prove that an isometry is injective is easy: For an isometry: ||f(x)-f(y)||=||x-y|| If x\neq y then ||x-y||>0 and therefore...
35. ### Trying to find an isometry T(X)=MX

Hi, I have a question: I am trying to find an isometry such that T(aU+bV)≠aT(U)+bT(V). I have tried so many possibilities. I gave T(X)=MX given that M is a matrix that doesn't have an inverse. But i still can't find a nice matrix that will make the proposition possible. Help please
36. ### Isometry from R to itself.

Homework Statement Find all isometries from the reals to itself. Homework Equations Well what we're basically doing is trying to find functions f: R -> R such that for any x, y in R, the property |f(x) - f(y)| = |x-y| holds. The Attempt at a Solution OK, so this shouldn't be too hard. It seems...
37. ### Fourier transform isometry

let S(R) be the schwartz space, M(R) be the set of moderately decreasing functions, F be the Fourier transform Suppose F:S(R)->S(R) is an isometry, ie is satisfies ||F(g)|| = ||g|| for every g in S(R). How is it possible that there exists a unique extension G: M(R)->M(R) which is an...
38. ### Isometry problem, pls help

Homework Statement Let (K, d) and (K', d') be two compact metric spaces and let f:K-->K' and g:K'--->K be isometries. Show that f(K)=K' and g(K')=K Homework Equations n/a The Attempt at a Solution I know that isometry implies that I can find one-to-one correspondence mapping, but...
39. ### Is an isometry always a statement about the principle of covariance?

Can someone help me out with this idea? Let's say we have a diffeomorphism. We know that under certain circumstances (invariance of the metric) this diffeomorphism is an isometry. Here is the part I'm not sure about. Is an isometry always just a statement about the principle of covariance...
40. ### Isometry and orbit dimensions

Hi people, I just need to verify that I understand this correctly. For some four dimensional manifold and group of isometries: the dimension of the isometry group is given by the number of Killing vectors, while the dimension of the orbit group is given by the number of linearly...
41. ### Can every isometry in the plane be composed of at most three reflections?

I need to show that every isometry in the plane is a composition of at most three reflections. Now every isometry in the plane is one of the four: 1.translation. 2.rotation. 3.reflection. 4.glide. now obviously refelction is itself. now for translation i showed with a picture that we...
42. ### Prove Isometry: Exists Orthonormal Basis V w/ \|Se_j\|=1

Homework Statement Prove or give a counterexample: if \mathcal{S} \in \mathcal{L} \left( V \right) and there exists and orthonormal basis \left( e_{1} , \ldots , e_{n} \right) of V such that \left\| \mathcal{S} e_{j} \right\| = 1 for each e_{j}, then \mathcal{S} is an isometry.Homework...
43. ### How to prove a function is an isometry

I need to show a particular map f:M-->N is an isometry (globally). M,N are riemannian manifolds, p is a point on M. That is, I need to show: <u,v>_p = <df_p(u),df_p(v)>_f(p) (for all p and for all u,v in T_p(M)?) I think my problem is that I don't understand what this statement really...
44. ### Proving Isometry on Metric: How to Do It

How do I prove every isometry from R->R is of the form f(x)=a+-x , regardless of the metric? I know it has to do with considering d_1(0,x_1)=d_2(0, f(x_1)), but beyond that I am lost.
45. ### How would I prove that an isometry is one to one?

How would I prove that an isometry is one to one? General definition of isometry, A: <Ax,Ay> = <x,y> Where < , > is an inner product (scalar product, dot product, etc.) How do I prove A has to be one to one for this to work?
46. ### Isometry & Inverse: Does Every Isometry Have an Inverse?

Does every isometry have an inverse?
47. ### Proving Isometry Preserves Distance in R^3 with f(0)=0

Ok I know that isometries preserve distance and in order for a fn to be an isometry || f(u) - f(v) || = || u - v || and in this question it asks to prove prove that if an isometry satisfies f(0) = 0 then we have f(u) x f(v) = +- f(u x v) and what property of f determines the choice of...