Characterization of Clifford Torus in Three-Spheres

Kim, Dong-Soo; Kim, Young Ho; Qian, Jinhua

doi:10.3390/math8050718

Cited by 3 publications

(5 citation statements)

References 10 publications

(16 reference statements)

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“…These manifolds will now be briefly explored, giving three different ways to interpret the emergence of spacetime. The first interpretation concerns the Clifford torus [66][67][68][69] and the parallelizable manifold S 3 × S 1 which is homeomorphic to M [36]. The second interpretation concerns de Sitter space [16,30,70].…”

Section: The Primordial Manifoldsmentioning

confidence: 99%

“…Giving each of these 1-spheres a radius: r = 1 / √ 2 , taking the Cartesian product of the two independent spaces produces the space [66][67][68][69]:…”

Section: The First Interpretation: Compactified Time-mmentioning

confidence: 99%

“…as a smooth topological manifold generated by the T GA. The T manifold is the Clifford torus [66][67][68][69], of which the standard torus in R 3 is an asymmetric projection [67]. As defined, the distance of any point on T to the origin of…”

Section: The First Interpretation: Compactified Time-mmentioning

confidence: 99%

“…Thus T can be perfectly embedded within S 3 , and is also a maximally symmetric minimal surface within S 3 which is the standard genus one splitting of S 3 [66][67][68][69].…”

Section: The First Interpretation: Compactified Time-mmentioning

confidence: 99%

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Quantum gravity through geometric algebra

Wolk

2023

J. Phys. A: Math. Theor.

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Geometric Algebra is used to construct a model of quantum gravity in which the 4D spacetime manifold is built from qubitic quantum entangled spacetime elements of geometric algebraic nature. This is achieved by first investigating the spontaneous decomposition of the spacetime continuum’s metric structure at Planck distances as one approaches what otherwise would become the Universe’s initial singularity. Considering this metric deconstruction within the model one finds that the reverse induced emergence of the spacetime continuum is generated by a coupling action triggered by the embedding action between distinct pre-spacetime manifolds. Further uncovered in this re-engineering of spacetime is a Poincaré-constrained quantum foam existing at infra-Planck scales.

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Section: The Primordial Manifoldsmentioning

confidence: 99%

“…Giving each of these 1-spheres a radius: r = 1 / √ 2 , taking the Cartesian product of the two independent spaces produces the space [66][67][68][69]:…”

Section: The First Interpretation: Compactified Time-mmentioning

confidence: 99%

Section: The First Interpretation: Compactified Time-mmentioning

confidence: 99%

“…Thus T can be perfectly embedded within S 3 , and is also a maximally symmetric minimal surface within S 3 which is the standard genus one splitting of S 3 [66][67][68][69].…”

Section: The First Interpretation: Compactified Time-mmentioning

confidence: 99%

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Quantum gravity through geometric algebra

Wolk

2023

J. Phys. A: Math. Theor.

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“…The most straightforward manner is to embed each of both circles composing a 2D-torus in an independent R 2 space, thus resulting that both circles need an R 2 × R 2 := R 4 space to be displayed. This representation is called Clifford torus [4], and it happens that if the radius of both circles is diagram where circles S a and S b , with radii r a and r b , are shown independently, as viewed in Figure 1a. However, the usual way to go is to get a stereographic projection in 3D of the torus in 4D, such as that depicted in Figure 1b, where a simple rotation in R 4 is projected into R 3 .…”

Section: Introductionmentioning

confidence: 99%

Arithmetic modeling of k-ary n-cubes and toroidal k-ary grids

Roig,

Alcaraz,

Gilly

et al. 2024

J. Phys.: Conf. Ser.

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The use of toroidal topologies offers many advantages in the computing field when it comes to both pattern spotting and path finding strategies. The former is covered by de Bruijn shapes, which permit to uniquely locate a single pattern throughout the shape. However, the latter is mainly carried out by k-ary n-cubes, which label node identifiers in a sequential order according to rows, columns, layers, and so on. This scheme facilitates the movement among nodes by just applying arithmetic operations, such as integer divisions and arithmetic modulo n. On the othe hand, toroidal k-ary grids are an alternative available in some specific cases, where determined patterns appear in all dimensions of each node, thus allowing to use those patterns to dictate paths to move among nodes. In this paper, the arithmetic bases of both path finding strategies have been presented and some pseudocode algorithms have been designed.

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Spheres and Tori as Elliptic Linear Weingarten Surfaces

Kim

Qian

2022

Mathematics

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The linear Weingarten condition with ellipticity for the mean curvature and the extrinsic Gaussian curvature on a surface in the three-sphere can define a Riemannian metric which is called the elliptic linear Weingarten metric. We established some local characterizations of the round spheres and the tori immersed in the 3-dimensional unit sphere, along with the Laplace operator, the spherical Gauss map and the Gauss map associated with the elliptic linear Weingarten metric.

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Characterization of Clifford Torus in Three-Spheres

Cited by 3 publications

References 10 publications

Quantum gravity through geometric algebra

Quantum gravity through geometric algebra

Arithmetic modeling of k-ary n-cubes and toroidal k-ary grids

Spheres and Tori as Elliptic Linear Weingarten Surfaces

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