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Rotations in differential geometry

  1. Jan 17, 2016 #1
    Simple and basic question(maybe not). How are rotations performed in differential geometry ?

    What does the rotation matrix look like in differential geometry? Let us assume we have orthogonal set of basis vectors initially.

    I am looking to calculate the angle between two geodesics. Can this local angle be related to the tangent vectors ?
     
  2. jcsd
  3. Jan 17, 2016 #2

    WWGD

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    If I understood you correctly, the angle between (differentiable) curves at a given point is defined as the angle between their tangents at the common point.
     
  4. Jan 17, 2016 #3
    Excellent. So if the two curves are geodesics then the "local angle" could be defined in terms of latitude and longitude. Then the tangent vectors could be defined in terms of latitude and longitude. Am I correct ?
     
  5. Jan 17, 2016 #4

    WWGD

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    Sorry, I don't get the reference to latitude and longitude outside of spheres. Would you elaborate?
     
  6. Jan 17, 2016 #5
    I am assuming that two curves are arcs of two great circles on the surface of the sphere with unit radius. That is assuming the earth is a perfect sphere and not an ellipsoid.
     
  7. Jan 18, 2016 #6

    Fredrik

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    The angle ##\theta## between two vectors x and y in a real inner product space is defined by ##\cos\theta =\frac{\langle x,y\rangle}{\|x\|\|y\|}##.

    If your two geodesics intersect at a point p in the manifold M, then their tangents (velocity vectors) are elements of ##T_pM##, i.e. the tangent space at p. If M is a Riemannian manifold, then ##T_pM## is a real inner product space.
     
  8. Jan 18, 2016 #7
    Thanks for that answer. But I am not a mathematician and I am dealing with Euclidean space in three dimensions - latitude, longitude and r (radius of uniform sphere). I am trying to correlate the mathematical definitions to definitions in cartography- differential geodesy. So if I have a geodesic pointing to true north and I have another arc which is east of that geodesic we call that angle between two arcs as the azimuth. So the tangent vectors of the two geodesics would give me the angle that I require ?
     
  9. Jan 22, 2016 #8
    Rotation
    Rotation seems to be a special concepts for three dimensional Euclidean space. But in a more general manifold orientation using n-form is usually used...
     
  10. Jan 24, 2016 #9
    (No: n-dimensional differential forms, and rotations, are two different concepts. )

    A rotation of n-dimensional Euclidean space, for any n ≥ 1, is a mapping

    f: nn

    of the space to itself such that

    a) f(0) = 0;

    b) all distances are preserved:

    ||f(x) - f(y)|| = ||x - y||​

    for all points x, y of n;

    and

    c) there is a continuous family {ft | 0 <= t <= 1} of mappings

    ft: n → ℝn

    satisfying a) and b) such that f0 = f and f1 = the identity mapping (i.e., taking each point of n to itself).

    * * *

    This definition given by a), b), and c) above is a conceptual definition.

    But it turns out to be very useful that it is equivalent to the following computational definition:

    f: nn

    is a rotation according to a), b), and c) if and only if f is an invertible linear mapping, given by an n × n matrix L satisfying

    i) L-1 = Lt

    (i.e., the inverse of L equals the transpose of L) and

    ii) det(L) > 0.

    (In fact when i) and ii) hold, then det(L) = +1, necessarily.)
     
  11. Jan 24, 2016 #10
    I do not understand why it should be restricted to Euclidean space.
    I am trying to use derivatives to describe rotations so that it can be linked to curvature eventually. If you have a conformal map then the transformation can be described in terms of a scaling term times the Jacobian matrix. Is that correct ? Let us assume I have two curves(geodesics) defined in terms of latitude and longitude on the surface of a sphere(or an ellipsoid). The first curve is defined with respect to one set of axes and the second curve is defined with respect to a rotated set of axes.

    Assuming I have these two functions
    λ' = f(λ,Φ)
    and Φ' = f (λ,Φ)

    Then I can define these basis vectors ## e' = ∑ P_i e_n ##
    between the unrotated and rotated system.
    I am going to call δ the local angle formed between two meridians at the point P.
     
  12. Jan 24, 2016 #11
    There are a few things I don't understand.

    You write: "If you have a conformal map then the transformation can be described in terms of a scaling term times the Jacobian matrix. Is that correct ?"

    1) I have to ask: What transformation do you mean by "the" transformation ?

    1a) What do you mean by "the" Jacobian matrix? Which Jacobian matrix?

    2) Where you write:

    "Assuming I have these two functions λ' = f(λ,Φ) and Φ' = f (λ,Φ)",

    aren't you making λ' and Φ' equal to each other, since you have set each of them equal to the same expression, f(λ,Φ) ?

    3) What do λ, λ', Φ, Φ' have to do with the two curves that you mention? Or with the conformal map you started with?

    4) Where you write: "Then I can define these basis vectors . . . between the unrotated and rotated system," this is the first time you mention any rotation. Which rotation?
     
  13. Jan 24, 2016 #12
    1) Regarding my usage of the word transformation - I have a curve defined with respect to a (λ,Φ) system. Then I rotate the axes of this system and define another curve in the rotated system with respect to (λ',Φ') system. Would that mapping be conformal ?

    2) Sorry there was an error in the transformation functions

    λ' = f(λ,Φ)

    and Φ' = g(λ,Φ)

    3) and 4) follow from (1). Please let me know if you need any clarification.
     
  14. Jan 24, 2016 #13
    If I had an object rotating around the earth(an imperfect ellipsoid) in a latitude circle or a great circle what sort of motion would that be ?
     
  15. Jan 25, 2016 #14
    I presume f and g would have invertible mappings from the definition that you outlined.
     
  16. Jan 25, 2016 #15

    WWGD

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    Sorry for getting back so late: If the two geodesics intersect at a point p, then the angle between the curves is defined as the angle between the tangent spaces.

    An issue with your reference to conformal maps is that these maps are defined on a sphere, instead of on Euclidean n-space ( or complex plane) , so you need to use some additional machinery (pre-and post- compose with charts, chart-maps) to define conformal maps on a sphere ( or in a general manifold) that you do not need to use in Euclidean space.
     
  17. Jan 25, 2016 #16
    Thanks for that note. Things are becoming clear now.
    I realized later I was not on a Euclidean space.
    So the tangent vectors can also be called the basis vectors of the transformation. Am I right ?

    So if each basis vector is defined in the following way

    ## e_i' = ∑ P_i e_i ##
    and given two transformation functions f and g - does the matrix of partial derivatives give me the matrix P ? I forgot to mention that f and g are conformal maps.
     
  18. Jan 25, 2016 #17
    "Sorry for getting back so late: If the two geodesics intersect at a point p, then the angle between the curves is defined as the angle between the tangent spaces."

    This is almost true. Actually, the angle between two parametrized curves that intersect at a point p where neither of them has zero velocity is the angle between their tangent vectors. (And for the sign of the angle to be well-defined, there should be an order assigned to the curves: a first curve and a second curve.)

    For instance: If in the plane, curve

    c(s) = (s,0)​

    for s ≥ 0, and curve

    d(t) = (1+2t,t)​

    for t ≥ 0, then they intersect at the point p = (1,0) with tangent vectors c'(1) = (1,0) and d'(0) = (2,1).

    Hence the angle θ between curve c and curve d at point p satisfies

    cos(θ) = (1,0)/1 (2,1)/√5 = 2/√5 = √(4/5).

    And, θ assumed taken counterclockwise from curve c to curve d will be a positive angle with 0 < θ < π/2.

    (If we had just used the tangent spaces, there would be an ambiguity not just with the sign of θ but worse, whether we meant θ or its supplement π-θ.)
     
  19. Jan 25, 2016 #18

    WWGD

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    So you specify ahead of time which of the angles you are interested in.
     
  20. Jan 25, 2016 #19

    WWGD

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    If your transformation is only a rotation, then yes, this is a conformal map. And I think you are considering Mobius maps , i.e., maps
    of the type:

    ## \frac {az+b} {cz+d} ## (with ## ad \neq bc ##) , then, yes, you can use standard polynomial division to express them as a composition of inversions, rotations and scalings (maybe some other transformation I cannot remember now) And all of these are bijective conformal maps from the sphere ##\mathbb CP^1 =S^2 ## to itself, where = is loosely used to mean manifold-isomorphic.
     
  21. Jan 25, 2016 #20
    I am just considering pure rotations at the moment. So I just calculate the matrix of partial derivatives right ?
     
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