# Name this 4D Manifold?

by Sorento7
Tags: manifold
 P: 16 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?
 Sci Advisor HW Helper PF Gold P: 4,771 I don't think so.
 Sci Advisor HW Helper PF Gold P: 2,602 If you view the 2-sphere as a complex projective space, then $\mathbb{P}^1\times \mathbb{P^1}$ is the 0th Hirzebruch surface, as well as the exceptional del Pezzo surface.
 Sci Advisor HW Helper PF Gold P: 4,771 Name this 4D Manifold? 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.
 Sci Advisor HW Helper P: 9,488 fred.
HW Helper
PF Gold
P: 2,602
 Quote by quasar987 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.
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 $S^2\times S^2$ is a Lagrangian submanifold of $\mathbb{C}^3$. He hasn't specified the other angle, but it can be chosen to give a $U(1)$ bundle over $S^2\times S^2$ with a connection 1-form that is

$$A_\psi = p \cos\theta_1 d\phi_1 + q \cos\theta_2 d\phi_2.$$

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

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

It turns out that $T^{1,1}$, viewed as the base of the conical metric on $\mathbb{C}^3$, is compatible with the Kahler structure on $\mathbb{C}^3$ The metric on $T^{1,1}$ 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 $\mathbb{C^3/Z_3}$ 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 $\mathcal{O}(-1)\oplus \mathcal{O}(-1) \rightarrow \mathbb{P^1}$.
 Sci Advisor HW Helper PF Gold P: 4,771 Interesting! Thanks. Is a conifold defined as just an orbifold with 1 singular pt?
$$ds^2 \sim dr^2 + r^2 d\Omega^2.$$