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The hyperriemann sphere

  1. Feb 4, 2016 #1
    As many of you know, using the stereographic projection one can construct a homeomorphism between the the complex plane ℂ1 and the unit sphere S2∈ℝ3. But the stereographic projection can be extended to
    the n-sphere/n-dimensional Euclidean space ∀n≥1. Now what I am talking about is the the mapping I: H→ H where H is the space of all Quaternions and I(q) = (1/q) ∀q ∈ H. So this complex manifold is the one-point compactification of H which I will refer to as ◊.
    That is, ◊ : H ∪ {∞}. I: 1/(0+0i+0j+0k) ↔ {∞}. So is there an official name for ◊ and has it already been shown that it is topologically equivalent to S4? I assume so but if need be I will give a proof attempt in a followup post.
     
  2. jcsd
  3. Feb 4, 2016 #2

    lavinia

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    While it is true that ##S^4## is the 1 point compactification of ##R^4##, it is not a complex manifold. In fact the tangent bundle does not even have an almost complex structure. I am not sure if any sphere other than the Riemann sphere can be a complex manifold. That said, the Riemann sphere is a natural extension of the complex plane for the study of complex analysis. You might want to research whether quaternionic analysis naturally extends to the 4 sphere. What about octonian analysis?
     
  4. Feb 5, 2016 #3

    My bad. It is a manifold but not a complex manifold. Now I don't know about Octonion analysis, however, the unit Octonions do not form a group as they are non-associative whereas the quaternions are a non-commutative group under multiplication and the unit pure Quaternions are isomorphic to SO(3).
     
  5. Feb 5, 2016 #4

    lavinia

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    Right.

    Maybe this is interesting.

    http://projecteuclid.org/download/pdf_1/euclid.bbms/1102715140
     
  6. Feb 5, 2016 #5
    Lavinia wrote: "I am not sure if any sphere other than the Riemann sphere can be a complex manifold."

    Neither is anyone else! It is known that the only other sphere Sn besides S2 that even has an almost complex structure . . .

    (((
    i.e., a bundle isomorphism

    J: T(Sn) →T(Sn)​

    such that

    J2 = -I,​

    where

    I: T(Sn) → T(Sn)
    is the identity. In this sense applying J to T(Sn) is "almost" like multiplication by the imaginary unit i applied to the complex plane ℂ.
    )))

    . . . is S6. (This comes from the fact that S6 is the underlying topological space of the "pure imaginary" unit octonions.)

    But it remains unknown whether S6 admits the structure of a complex analytic manifold.

    --------------------------------------------------------------------

    EinsteinKreuz wrote ". . . the unit pure Quaternions are isomorphic to SO(3)."

    Close, but not quite. The unit-length quaternions form the unit sphere in 4-space, known as S3. This is the unique double covering space of the rotation group SO(3) of 3-space. In fact, the underlying topological space of SO(3) is 3-dimensional (real) projective space, P3, which is obtained by identifying antipodal points of S3.
     
    Last edited: Feb 5, 2016
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