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Clifford Torus


The Clifford torus is the product of two circles of radius 1/sqrt(2) in the unit sphere S^3 subset R^4. A parametrization is

 X(u,v)=2^(-1/2)(cosu,sinu,cosv,sinv),

where 0<=u,v<2pi. Its induced metric is flat, its mean curvature vector vanishes, and its area is 2pi^2. It is a minimal Clifford hypersurface, but is not a totally geodesic submanifold. It gives the dimension-two comparison value in the Solomon-Yau least-volume conjecture.


See also

Minimal Clifford Hypersurface, Minimal Submanifold, Solomon-Yau Least-Volume Conjecture, Torus

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References

Ge, J. and Li, F. "Volume Gap for Minimal Submanifolds in Spheres." J. Geom. Anal. 36, Article No. 291, 2026. https://doi.org/10.1007/s12220-026-02551-5.

Cite this as:

Weisstein, Eric W. "Clifford Torus." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CliffordTorus.html

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