arXiv research

Study physical work done by isotropic vector forces along isotropic curves.

Killing vector fields of a closed homogeneous and isotropic universe are studied. It is shown that in general case there is no time-like Killing vector fields in such a universe. Two exceptional cases are revealed.

Geodesic completeness and flow properties of compact Brinkmann spacetimes proven.

Aguilar introduced isotropic almost complex structures $J_{δ, σ}$ on the tangent bundle of a Riemannian manifold $(M, g)$. In this paper, some results will be obtained on the integrability of these structures. These structures with the Liouville 1-form define a class of Riemannian metrics $g_{δ, σ}$ on $T M$ which are …

Germs of tubular neighborhood embeddings for submanifolds N of manifolds M are in one-one correspondence with germs of Euler-like vector fields near N. In many contexts, this reduces the proof of `normal forms results' for geometric structures to the construction of an Euler-like vector field compatible with the given …

It is established that the existence of non-isotropic vector field which Jacobi operator of maximal rank is an obstacle for the existence of non-trivial second-order symmetric parallel tensor field. In turns out that presence of such obstacle follows that manifold as pseudo-Riemannian manifold is locally non-reducible.…

We study the behaviour of differential forms in a manifold having at least one of their maximal isotropic local distributions endowed with the special algebraic property of being decomposable. We show that they can be represented as the sum of a form with constant coefficients and one that vanishes whenever contracted …

Study projective symmetries in Finsler spaces, showing reductions and constant flag curvature.

Study compact plane waves, showing they are essentially standard.

The study of Bonnet surfaces in 4D space forms reveals new conformally invariant properties and characterizes proper Bonnet surfaces.

The isotropic almost complex structures induce a Riemannian metric $g_{δ,σ}$ on TM, which are the generalized type of Sasakian metric. In this paper, the Levi-Civita connection of $g_{δ,σ}$ is calculated and the harmonicity of unit vector fields from $(M, g)$ to $(S(M), i*g_{δ,0})$ is investigated, where $i*g_{δ,0}$ is…

Classifies left invariant Kundt structures on 3D Lie groups.

The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian…

Study timelike surfaces with parallel mean curvature in Minkowski 4-space.

Study isotropic solutions in smooth metric measure spaces with vacuum Einstein equations.

Let $M$ be a complete Riemannian manifold and suppose $p\in M$. For each unit vector $v \in T_p M$, the $\textit{Jacobi operator}$, $\mathcal{J}_v: v^\perp \rightarrow v^\perp$ is the symmetric endomorphism, $\mathcal{J}_v(w) = R(w,v)v$. Then $p$ is an $\textit{isotropic point}$ if there exists a constant $κ_p \in \mat…

Study loxodromes and geodesics on rotational surfaces in pseudo-isotropic space.

Paper calculates KL divergence for isotropic Gaussian-Markov fields.

We introduce the notion of commuting Ricci tensor for real hypersurfaces in the complex quadric $Q^m = SO_{m+2}/SO_mSO_2$ . It is shown that the commuting Ricci tensor gives that the unit normal vector field $N$ becomes $\frak A$-principal or $\frak A$-isotropic. Then according to each case, we give a complete classifi…

In this paper, we first deduce a formula of S-curvature of homogeneous Finsler spaces in terms of Killing vector fields. Then we prove that a homogeneous Finsler space has isotropic S-curvature if and only if it has vanishing S-curvature. In the special case that the homogeneous Finsler space is a Randers space, we giv…

Constructs a Morse-Bott function on symplectic Grassmannians.

Continuous representation of words is a standard component in deep learning-based NLP models. However, representing a large vocabulary requires significant memory, which can cause problems, particularly on resource-constrained platforms. Therefore, in this paper we propose an isotropic iterative quantization (IIQ) appr…

We consider 3D flow equations inspired by the renormalization group (RG) equations of string theory with a three dimensional target space. By modifying the flow equations to include a U(1) gauge field, and adding carefully chosen De Turck terms, we are able to extend recent 2D results of Bakas to the case of a 3D Riema…

A semi-isotropic space is a real affine 3-space endowed with the non-degenerate metric dx^{2}-dy^{2}. The main purpose of this paper is to describe the surfaces of revolution in the semi-isotropic space that satisfy some equations in terms of the position vector and the Laplace operators with respect to the first and t…

In this paper, we show that isotropic Lagrangian submanifolds in a $6$-dimensional strict nearly Kähler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the $J$-isotropic Lagrangian submanifolds in the homogeneous nearly Kähler $\mathbb{S}^3\times \mathbb{S}^3$ is also…

The paper proves the existence of H-spheres with arbitrary codimensions in certain Riemannian manifolds.

We will investigate the local geometry of the surfaces in the $7$-dimensional Euclidean space associated to harmonic maps from a Riemann surface $Σ$ into $S^6$. By applying methods based on the use of harmonic sequences, we will characterize the conformal harmonic immersions $\varphi:Σ\to S^6$ whose associated immersio…

The closed homogeneous and isotropic universe is considered. The bundles of Weyl and Dirac spinors for this universe are explicitly described. Some explicit formulas for the basic fields and for the connection components in stereographic and in spherical coordinates are presented.

A spinorial approach to 6-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the 4-dimensional case, making full use of the SO(6) symmetry to uncover results not easily seen in the tensorial approach. U…

Consider a symplectic manifold $M$, a Hamiltonian vector field $X$ and a fibration $Π:M\rightarrow N$. Related to these data we have a generalized version of the (time-independent) Hamilton-Jacobi equation: the $Π$-HJE for $X$, whose unknown is a section $σ:N\rightarrow M$ of $Π$. The standard HJE is obtained when the …

Let $R^{n+1, n}$ be the vector space $R^{2n+1}$ equipped with the bilinear form $(X,Y)=X^t C_n Y$ of index $n$, where $C_n= \sum_{i=1}^{2n+1} (-1)^{n+i-1} e_{i, 2n+2-i}$. A smooth $γ: R\to R^{n+1,n}$ is {\it isotropic} if $γ, γ_x, \ldots, γ_x^{(2n)}$ are linearly independent and the span of $γ, \ldots, γ_x^{(n-1)}$ is …

We prove ultradifferentiable Chevelley restriction theorems for a wide range of ultradifferentiable classes. As a special case we find that isotropic functions, i.e., functions defined on the vector space of real symmetric matrices invariant under the action of the special orthogonal group by conjugation, possess some …

The paper examines isotropic cosmological space-times with changing sectional curvature.

A complete classification of isotropic vector equations of the geometric type that possess higher symmetries is proposed. New examples of integrable multi-component systems of the geometric type and their auto-Backlund transformations are found.

The paper analyzes how grid cells perform path integration and learns hexagon grid patterns.

The isotropic 3-space \mathbb{I}^{3} is a real affine 3-space endowed with the metric dx^{2}+dy^{2}. In this paper we describe Weingarten and linear Weingarten affine translation surfaces in \mathbb{I}^{3}. Further we classify the affine translation surfaces in \mathbb{I}^{3} that satisfy certain equations in terms of …

The concept of pure spinor is generalized, giving rise to the notion of pure subspaces, spinorial subspaces associated to isotropic vector subspaces of non-maximal dimension. Several algebraic identities concerning the pure subspaces are proved here, as well as some differential results. Furthermore, the freedom in the…

We introduce the notion of $\varepsilonη\,$-Einstein $\varepsilon\,$-contact metric three-manifold, which includes as particular cases $η\,$-Einstein Riemannian and Lorentzian (para) contact metric three-manifolds, but which in addition allows for the Reeb vector field to be null. We prove that the product of an $\vare…

New spectral mixture representation for isotropic kernels simplifies random Fourier features.

Bayesian Neural ODEs improve vessel trajectory prediction with better uncertainty estimates.

The paper finds a geometric way to minimize a convex function to construct isotropic measures.

AniDS improves molecular force field modeling by learning anisotropic noise.

Isotropic kernels' performance is analyzed across different tasks with and without invariants.

Study minimal rational curves on complex manifolds with isotropic VMRT.

Overview of marginally trapped surfaces in various spacetimes.

The geometrical structures (in the sense of E. Cartan) are analyzed which underlie the gravitational radiation phenomenon. Among the results are : - the introduction of the adapted frame bundle to a congruence of isotropic hypersurfaces in a Lorentzian manifold, - the description of the reduced frame bundle which admit…

Study on InstaHide's security, linking to phase retrieval problem.

In this paper we study the geometry of manifolds with vector cross product and its complexification. First we develop the theory of instantons and branes and study their deformations. For example they are (i) holomorphic curves and Lagrangian submanifolds in symplectic manifolds and (ii) associative submanifolds and co…