arXiv research

Natural volume forms defined for pseudo-Finslerian manifolds with specific metrics.

The nonzero level sets of a homogeneous, logarithmically homogeneous, or translationally homogeneous function are affine spheres if and only if the Hessian determinant of the function is a multiple of a power or an exponential of the function. In particular, the nonzero level sets of a homogeneous polynomial are proper…

Deformations of singular Cayley submanifolds studied.

In this article we study the deformation theory of conically singular Cayley submanifolds. In particular, we prove a result on the expected dimension of a moduli space of Cayley deformations of a conically singular Cayley submanifold. Moreover, when the Cayley submanifold is a two-dimensional complex submanifold of a C…

Desingularizes conically singular Cayley submanifolds.

Third in a series, this paper constructs non-trivial Cayley fibrations with conical singularities.

The study examines discrete curvature notions on Cayley graphs of certain groups.

Study Cayley fibrations on Bryant-Salamon $Spin(7)$ manifolds.

Given a Kaehler manifold of complex dimension 4, we consider submanifolds of (real) dimension 4, whose Kaehler angles coincide. We call these submanifolds Cayley. We investigate some of their basic properties, and prove that (a) if the ambient manifold is a Calabi-Yau, the minimal Cayley submanifolds are just the Cayle…

Eastwood and Ezhov generalized the Cayley surface to the Cayley hypersurface in each dimension, proved some characteristic properties of the Cayley hypersurface and conjectured that a homogeneous hypersurface in affine space satisfying these properties must be the Cayley hypersurface. We will prove this conjecture when…

Strictly enforcing orthonormality constraints on parameter matrices has been shown advantageous in deep learning. This amounts to Riemannian optimization on the Stiefel manifold, which, however, is computationally expensive. To address this challenge, we present two main contributions: (1) A new efficient retraction ma…

The study introduces Cayley--Abels--Rosendal graphs for Polish groups.

Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.

We define Cayley structures as a field of Cayley's ruled cubic surfaces over a four dimensional manifold and motivate their study by showing their similarity to indefinite conformal structures and their link to differential equations. In particular, for Cayley structures an extension of certain notions defined for inde…

Cayley cones in the octonions $\mathbb{O}$ that are ruled by oriented 2-planes are equivalent to pseudoholomorphic curves in the Grassmannian of oriented 2-planes G(2,8). The well known twistor fibration $G(2,8) -> S^6$ is used to prove the existence of immersed higher-genus pseudoholomorphic curves in $\gro$. Equivale…

The Cayley graph of quandles reveals structural properties and is studied for various classes.

Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras $\mathfrak{f}_4$ and $\mathfrak{f}^*_4$, i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we cla…

In this paper we study almost complex and almost para-complex Cayley structures on six-dimensional pseudo-Riemannian spheres in the space of purely imaginary octaves of the split Cayley algebra $\mathbf{Ca}'$. It is shown that the Cayley structures are non-integrable, their basic geometric characteristics are calculate…

Introduces Lorentzian Cayley form solving geometric puzzle.

Study shows only two topological configurations for Spin(7)-manifold fibrations, ruling out smooth Cayley fibrations.

Let $X$ be a smooth irreducible nondegenerate projective variety and let $X^*$ denote its dual variety. It is well known that $σ_2(X)^*$, the dual of the 2-secant variety of $X$, is a component of the singular locus of $X^*$. The locus of bitangent hyperplanes, i.e. hyperplanes tangent to at least two points of $X$, is…

Functions of several octonion variables are investigated and integral representation theorems for them are proved. With the help of them solutions of the ${\tilde {\partial}}$-equations are studied. More generally functions of several Cayley-Dickson variables are considered. Integral formulas of the Martinelli-Bochner,…

We exhibit a family of homogeneous hypersurfaces in affine space, one in each dimension, generalising the Cayley surface.

The paper studies properties of Artin monoid Cayley graphs and their quasi-isometry to Deligne complexes.

Minkowski's second theorem can be stated as an inequality for $n$-dimensional flat Finsler tori relating the volume and the minimal product of the lengths of closed geodesics which form a homology basis. In this paper we show how this fundamental result can be promoted to a principle holding for a larger class of Finsl…

Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) to study the comass of certain self-dual 4-forms on R^8. In particular, we prove that the Cayley 4-form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio is SO(8)-conjugate to the Cayley 4-form. We also…

We prove an existence theorem for Spin(7)-instantons, which are highly concentrated near a Cayley submanifold; thus giving a partial converse to Tian's foundational compactness theorem. As an application, we show how to construct Spin(7)-instantons on Spin(7)-manifolds with suitable local K3 Cayley fibrations. This rec…

The Cayley hyperbolic space minimizes volume entropy among finite-volume metrics.

This paper develops graph theory for racks and quasigroups.

Groups can act on 3-manifolds if their Cayley complex can embed in specific types of 3-manifolds.

In this article, we consider Cayley deformations of a compact complex surface in a Calabi--Yau four-fold. We will study complex deformations of compact complex submanifolds of Calabi--Yau manifolds with a view to explaining why complex and Cayley deformations of a compact complex surface are the same. We in fact prove …

In this paper we introduce the area of 2-ruled 4-folds in R^n (n=7 or 8), that is, submanifolds M of R^n that admit a fibration over some 2-fold Sigma such that each fibre is an affine 2-plane in R^n. This is motivated by the paper math.DG/0012060 by Joyce on ruled special Lagrangian 3-folds in C^3 and the work of the …

We construct a Kaehler structure on the punctured cotangent bundle of the Cayley projective plane whose Kaehler form coincides with the natural symplectic form on the cotangent bundle and we show that the geodesic flow action is holomorphic and is expressed in a quite explicit form. We also give an embedding of the pun…

Upper bounds for Steklov eigenvalues in subgraphs of polynomial growth Cayley graphs.

This paper summarizes closed-form relations for SE(3) maps and their derivatives.

Cayley's (ruled cubic) surface carries a three-parameter family of twisted cubics. We describe the contact of higher order and the dual contact of higher order for these curves and show that there are three exceptional cases.

This paper shows that, away from 6, the kernel of the Witten genus is precisely the ideal consisting of (bordism classes of) Cayley plane bundles with connected structure group, but only after restricting the Witten genus to string bordism. It does so by showing that the divisibility properties of Cayley plane bundle c…

Open problem: Establishing bounds for Cayley-table completion to discover discrete algorithmic axioms.

We prove that the Cayley graph and the coset geometry of the von Dyck group $D(a,b,c)$ are linked by a vertex-to-edge duality.

Let M be an 8-manifold with a Spin(7)-structure. We first show that closed Cayley submanifolds of M form a smooth moduli space for a generic Spin(7)-structure. Then we study the deformations of a compact, connected Cayley submanifold X of M with non-empty boundary contained in a given submanifold W of M such that X and…

Investigates second-order conditions for Cayley forms in eight dimensions.

The result of the classical invariant theory (CIT) commonly referred to as Lemma of Cayley is reviewed and its analogue in the invariant theory of Killing tensors (ITKT) defined in pseudo-Riemannian spaces of constant curvature is formulated and proven. Illustrative examples are provided.

A nearly parallel $G_{2}$-manifold $Y$ is a Riemannian 7-manifold whose cone $C(Y) = \mathbb{R}_{>0} \times Y$ has the holonomy group contained in ${\rm Spin(7)}$. In other words, it is a spin 7-manifold with a real Killing spinor. We have a special class of calibrated submanifolds called Cayley submanifolds in $C(Y)$.…

The paper studies the graph geometry of finite groups, creating a dataset and analyzing its properties.

CTEF fits ellipsoids to noisy data in any dimension.

The study embeds infinite-dimensional geometric structures in Cayley graphs.

We study the deformations of an asymptotically cylindrical Cayley submanifold inside an asymptotically cylindrical Spin(7)-manifold. We prove an index formula for the operator of Dirac type that arises as the linearisation of the deformation map and show that if the Spin(7)-structure is generic, then there are no obstr…

We describe an action of $U(1)\times Sp(1)$ on $HP^7$ that allows to obtain a $Z_2$-quotient of CAYLEY (the Grassmannian of oriented 4-planes in $R^8$ closed under the three-fold cross product) by quaternionic Kähler reduction.