arXiv research

A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the sam distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous if for any two point $x_1, x_2\in M$ there is a Clifford-Wolf translation $ρ$ such that $ρ(x_1)=x_2$. In this paper, we study Clifford-Wolf transl…

The paper studies 4-qubit Clifford states and their properties.

The Clifford group for 2 qubits is divided into 20 orbits, each with 4608 matrices.

Classifies 4D spaces with compact Clifford-Klein forms.

We compute the structure groups of almost even-Clifford Hermitian manifolds and determine when such groups lead to Spin structures.

The paper studies Clifford-Bianchi groups acting on hyperbolic spaces and their properties.

We describe smooth compactifications of certain families of reductive homogeneous spaces such as group manifolds for classical Lie groups, or pseudo-Riemannian analogues of real hyperbolic spaces and their complex and quaternionic counterparts. We deduce compactifications of Clifford-Klein forms of these homogeneous sp…

We study manifolds endowed with an (almost) even Clifford (hermitian) structure and admitting a large automorphism group. We classify them when they are simply connected and the dimension of the automorphism group is maximal, and also prove a gap theorem for the dimension of the automorphism group.

Solves classification of compact Clifford-Klein forms for specific Lie groups.

Let $(M,F)$ be a connected Finsler space. An isometry of $(M,F)$ is called a Clifford-Wolf translation (or simply CW-translation) if it moves all points the same distance. The compact Finsler space $(M,F)$ is called restrictively Clifford-Wolf homogeneous (restrictively CW-homogeneous) if for any two sufficiently close…

Betten and Riesinger have shown that Clifford parallelism on real projective space is the only topological parallelism that is left invariant by a group of dimension at least 5. We improve the bound to 4. Examples of different parallelisms admitting a group of dimension 3 are known, so 3 is the "critical dimension".

Paper describes invariants of slice regular functions' automorphism group.

An almost Clifford and an almost Cliffordian manifold is a $G$--structure based on the definition of Clifford algebras. An almost Clifford manifold based on $\mathcal O:= \cc l (s,t)$ is given by a reduction of the structure group $GL(km, \mathbb R)$ to $GL(m, {\mathcal O})$, where $k=2^{s+t}$ and $m \in \mathbb N$. An…

Study geodesic orbit property on pseudo-Riemannian H-type nilmanifolds.

We determine the Lagrangian monodromy group L(T) and the smooth monodromy group S(T) of a Clifford torus T in the symplectic 4-space. We show that L(T) is isomorphic to the infinite dihedral group, and S(T) is generated by three reflections. We give explicit formulas for both groups. We also show that if a Lagrangian t…

We use a computer-aided approach to prove that there are no standard compact Clifford-Klein forms of homogeneous spaces of exceptional Lie groups. This yields further support for Kobayashi's conjecture about possible compact Clifford-Klein forms. On one hand, our approach is based on the algorithms developed in this wo…

The paper constructs metrics with non-negative curvature and harmonic maps.

Proves an equivariant version of index theorem for geometric families.

In this paper we study unitary braid group representations associated with Majorana Fermions. Majorana Fermions are represented by Majorana operators, elements of a Clifford algebra. The paper recalls and proves a general result about braid group representations associated with Clifford algebras, and compares this resu…

We obtain the topological obstructions to existence of a bundle of irreducible real Clifford modules over a pseudo-Riemannian manifold $(M,g)$ of arbitrary dimension and signature and prove that bundles of Clifford modules are associated to so-called real Lipschitz structures. The latter give a generalization of spin s…

In this article, we discuss the local rigidity of Clifford-Klein forms of homogeneous spaces of 1-connected completely solvable Lie groups. In fact, we introduce a splitting of the local rigidity: vertical rigidity and horizontal rigidity. By using this splitting, we refine some existing results about the local rigidit…

This work reconsiders the holomorphic and anti-holomorphic Dirac operators of Hermitian Clifford analysis to determine whether or not they are the natural generalization of the orthogonal Dirac operator to spaces with complex structure. We argue the generalized gradient construction of Stein and Weiss based on represen…

Study cohomology rings of Grassmannians using Clifford algebras and symmetric spaces.

Study minimizes CR surfaces in Heisenberg group with rotational symmetry.

Study on Einstein manifolds with specific properties.

We introduce the notion of even Clifford structures on Riemannian manifolds, a framework generalizing almost Hermitian and quaternion-Hermitian geometries. We give the complete classification of manifolds carrying parallel even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler …

In this paper, we introduce a new type of Finsler metrics, called $(α_1,α_2)$-metrics. We define the notion of the good datum of a homogeneous $(α_1,α_2)$-metric and use that to study the geometric properties. In particular, we give a formula of the S-curvature and deduce a condition for the S-curvature to be vanishing…

For each integer m>1 and l>0 we construct a pair of compact embedded minimal surfaces of genus 1+4m(m-1)l. These surfaces desingularize the m Clifford tori meeting each other along a great circle at the angle of π/m. They are invariant under a finite group of screw motions and have no reflection symmetry across a great…

New minimal surfaces desingularize three Clifford tori, proving uniqueness and characterizing them.

A generalized Clifford manifold is proposed in which there are coordinates not only for the basis vector generators, but for each element of the Clifford group, including the identity scalar. These new quantities are physically interpreted to represent internal structure of matter (e.g. classical or quantum spin). The …

Researchers found a counterexample disproving a 1962 conjecture.

Let $(M,F)$ be a connected Finsler space and $d$ the distance function of $(M,F)$. A Clifford translation is an isometry $ρ$ of $(M,F)$ of constant displacement, in other words such that $d(x,ρ(x))$ is a constant function on $M$. In this paper we consider a connected simply connected symmetric Finsler space and a discr…

Novel CG-EGNNs learn equivariant functions from Clifford algebras.

This paper has been withdrawn by the author due to a crucial sign error in equation 1. An isometry $ρ$ of a connected Finsler space $(M, F)$ is called bounded if the function $d(x, ρ(x))$ is bounded on $M$. It is called a Clifford-Wolf translation if the function $d(x, ρ(x))$ is constant on $M$. In this paper, we prove…

Betten and Riesinger constructed Parallelisms of $\mathop{\rm PG}(3,\mathbb R)$ with automorphism group $\mathop{\rm SO}(3,\mathbb R)$ by applying the reducible $\mathop{\rm SO}(3,\mathbb R)$-action to a rotational Betten spread. This was generalized by the present author so as to include oriented parallelisms (i.e., p…

In this paper, we continue the study of the existence problem of compact Clifford-Klein forms from a cohomological point of view, which was initiated by Kobayashi-Ono and extended by Benoist-Labourie and the author. We give an obstruction to the existence of compact Clifford-Klein forms by relating a natural homomorphi…

The paper examines smoothness in graded skew Clifford algebras.

In this paper we study a Clifford algebra generalization of the quaternions and its relationship with braid group representations related to Majorana fermions. The Fibonacci model for topological quantum computing is based on the fusion rules for a Majorana fermion. Majorana fermions can be seen not only in the structu…

Develops deformation theory for mapping spaces related to Morse theory and Bott-Thom isomorphism.

Study relates Finsler structures to Clifford bundles for flat metrics.

Extends Kostant's results to symmetric pairs in Clifford algebras.

A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the same distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous if for any two points $x_1, x_2\in M$ there is a Clifford-Wolf translation $ρ$ such that $ρ(x_1)=x_2$. In this paper, we give a complete classifi…

Class lecture notes at a beginning graduate level on the mathematical background needed to understand classical gauge theory. Covers group actions, fiber bundles, principal bundles, connections, gauge transformations, parallel transport, curvature, covariant derivatives, pseudo-riemannian manifolds, lagrangians, cliffo…

We introduce the concept of a Clifford-Weyl structure on a conformal manifold, which consists of an even Clifford structure parallel with respect to the tensor product of a metric connection on the Clifford bundle and a Weyl structure on the manifold. We show that the Weyl structure is necessarily closed except for som…

We introduce a notion of twisted pure spinor in order to characterize, in a unified way, all the special Riemannian holonomy groups just as a classical pure spinor characterizes the special Kähler holonomy. Motivated by certain curvature identities satisfied by manifolds admitting parallel twisted pure spinors, we also…

New symmetric Willmore tori emerge from Clifford torus in Berger spheres.

Extends spinor-horosphere correspondence to higher dimensions and new spinor types.

We consider the diffeological version of the Clifford algebra of a (diffeological) finite-dimensional vector space; we start by commenting on the notion of a diffeological algebra (which is the expected analogue of the usual one) and that of a diffeological module (also an expected counterpart of the usual notion). Aft…