Name this 4D Manifold?

  1. We know that a Clifford torus is parameterized in 4D euclidean space by:
    (x1,x2,x3,x4) = (Sin(theta1), Cos(theta1), Sin(theta2), Cos(theta2))
    {0<=theta1 and theta2<2pi}

    Consider that a clifford torus is the immediate result of Circle * Circle

    Now, have you encountered a similar manifold which is a result of Sphere * Sphere?
    The parameterization is quite straight-forward in 6 dimensions:
    (x1,x2,x3,x4,x5,x6)=(Sin(theta1)Cos(phi1), Cos(theta1)Cos(phi1), Sin(phi1), Sin(theta2)Cos(phi2), Cos(theta2)Cos(phi2), Sin(phi2))
    {0<= All angles< 2pi}

    Does there exist any name for this special 4D manifold?
     
  2. jcsd
  3. quasar987

    quasar987 4,770
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    I don't think so.
     
  4. fzero

    fzero 2,682
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    If you view the 2-sphere as a complex projective space, then [itex]\mathbb{P}^1\times \mathbb{P^1}[/itex] is the 0th Hirzebruch surface, as well as the exceptional del Pezzo surface.
     
  5. quasar987

    quasar987 4,770
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    I think the question is not so much "what are other names for S² x S²?" as it is "is there a name for the obvious embedding of S² x S² into R^6=R³ x R³?".

    Because that is what the Clifford torus is: it is just a name for the obvious embedding of S^1 x S^1 into R^4 = R² x R².
     
  6. mathwonk

    mathwonk 9,703
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    fred.
     
  7. fzero

    fzero 2,682
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    Ah, got it. As an aside, the range of angles quoted is wrong to get 2-spheres, but this is easily corrected. In fact there is a very beautiful structure associated with this type of embedding. I will mention a few highlights.

    This [itex]S^2\times S^2[/itex] is a Lagrangian submanifold of [itex]\mathbb{C}^3[/itex]. He hasn't specified the other angle, but it can be chosen to give a [itex]U(1)[/itex] bundle over [itex]S^2\times S^2[/itex] with a connection 1-form that is

    [tex] A_\psi = p \cos\theta_1 d\phi_1 + q \cos\theta_2 d\phi_2.[/tex]

    These bundles are known as [itex]T^{p,q}[/itex]. Two special examples are [itex]T^{0,1} = S^2\times S^3[/itex], which uses the Hopf fibration, while

    [tex]T^{1,1} = SU(2)\times SU(2)/U(1).[/tex]

    It turns out that [itex]T^{1,1}[/itex], viewed as the base of the conical metric on [itex]\mathbb{C}^3[/itex], is compatible with the Kahler structure on [itex]\mathbb{C}^3[/itex] The metric on [itex]T^{1,1}[/itex] can be chosen to be Einstein, which makes it a nontrivial example of an Einstein-Sasaki manifold.

    In fact, an explicit metric can be written down that describes the small resolution of the singularity in the orbifold [itex]\mathbb{C^3/Z_3}[/itex] viewed as a Calabi-Yau manifold (this is also known as the conifold singularity). The resolved conifold can also be viewed as the total space of the bundle [itex]\mathcal{O}(-1)\oplus \mathcal{O}(-1) \rightarrow \mathbb{P^1}[/itex].
     
  8. quasar987

    quasar987 4,770
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    Interesting! Thanks.

    Is a conifold defined as just an orbifold with 1 singular pt?
     
  9. fzero

    fzero 2,682
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    In this case, it is any singularity such that the metric in a small neighborhood of the singular point behaves like

    [tex] ds^2 \sim dr^2 + r^2 d\Omega^2.[/tex]

    There are probably some more technical mathematical details that, as a physicist, I will get wrong. The conifold is a local model for the behavior near isolated singularities of some larger manifold. I'm only familiar with the case where these singularities are describable as orbifold singularities of some neighborhood of the singular point.
     
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