MDPI - Publisher of Open Access Journals

64 pages, 661 KB

Open AccessReview

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 2 | Viewed by 4521

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 KB

Open AccessArticle

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 15 | Viewed by 5973

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.

C (K) = S^{3} \ (K \times 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)

► Show Figures

Figure 1

Search Results (2)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI

Saved Queries

Search Filter Reset All

Years

Feature Papers

Subjects

Journals

Article Types

Countries / Regions

Search Results (2)

Further Information

Guidelines

MDPI Initiatives

Follow MDPI