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Keywords = Thurston geometries

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64 pages, 661 KiB  
Review
A Comprehensive Survey on Parallel Submanifolds in Riemannian and Pseudo-Riemannian Manifolds
by Bang-Yen Chen
Axioms 2019, 8(4), 120; https://doi.org/10.3390/axioms8040120 - 30 Oct 2019
Cited by 1 | Viewed by 3893
Abstract
A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden–Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. [...] Read more.
A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden–Bortolotti connection. From submanifold point of view, parallel submanifolds are the simplest Riemannian submanifolds next to totally geodesic ones. Parallel submanifolds form an important class of Riemannian submanifolds since extrinsic invariants of a parallel submanifold do not vary from point to point. In this paper, we provide a comprehensive survey on this important class of submanifolds. Full article
(This article belongs to the Special Issue Geometric Analysis and Mathematical Physics)
26 pages, 1071 KiB  
Article
Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics
by Torsten Asselmeyer-Maluga
Symmetry 2019, 11(10), 1298; https://doi.org/10.3390/sym11101298 - 15 Oct 2019
Cited by 14 | Viewed by 5286
Abstract
In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement [...] Read more.
In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement C ( K ) = S 3 \ ( K × D 2 ) of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of S 3 branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk D 3 branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld–Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson–Thompson model). Full article
(This article belongs to the Special Issue Number Theory and Symmetry)
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12 pages, 2223 KiB  
Article
On Maximal Homogeneous 3-Geometries and Their Visualization
by Emil Molnár, István Prok and Jeno Szirmai
Universe 2017, 3(4), 83; https://doi.org/10.3390/universe3040083 - 4 Dec 2017
Cited by 3 | Viewed by 3815
Abstract
The motivation for this talk and paper is related to the classification of the homogeneous simply connected maximal 3-geometries (the so-called Thurston geometries: E 3 , S 3 , H 3 , S 2 × R , H 2 × R , [...] Read more.
The motivation for this talk and paper is related to the classification of the homogeneous simply connected maximal 3-geometries (the so-called Thurston geometries: E 3 , S 3 , H 3 , S 2 × R , H 2 × R , S L 2 R ˜ , Nil , and Sol ) and their applications in crystallography. The first author found in (Molnár 1997) (see also the more popular (Molnár et al. 2010; 2015) with co-author colleagues, together with more details) a unified projective interpretation for them in the sense of Felix Klein’s Erlangen Program: namely, each S of the above space geometries and its isometry group Isom ( S ) can be considered as a subspace of the projective 3-sphere: S P S 3 , where a special maximal group G = Isom ( S ) Coll ( P S 3 ) of collineations acts, leaving the above subspace S invariant. Vice-versa, we can start with the projective geometry, namely with the classification of Coll ( P S 3 ) through linear transforms of dual pairs of real 4-vector spaces ( V 4 , V 4 , R , ) = P S 3 (up to positive real multiplicative equivalence ∼) via Jordan normal forms. Then, we look for projective groups with 3 parameters, and with appropriate properties for convenient geometries described above and in this paper. Full article
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