arXiv research

An identity of conformal-projective curvature tensor of a statistical manifold is studied in this paper. The relation between the constancy of curvature and conformal-projective flatness of statistical manifolds is also discussed.

In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under certain condition, and derive the constancy of the principal curvatures of such submanifolds. As its result, we can derive that …

Paper studies curvature in Finsler geometry, proving curvature constancy under isotropy.

In this paper, the Cartan frames and the equi-affine curvatures are described with the help of the Frenet frames and the Frenet curvatures of a non-null and non-degenerate curve in a 3-dimensional pseudo-Riemannian manifold. The constancy of the Frenet curvatures of such a curve always implies the constancy of the equi…

Study almost rigidity of super Ricci flow with non-negative Muller quantity.

Tanno [6] provided an algebraic characterization in an almost Hermitian manifold to reduce to a space of constant holomorphic sectional curvature, which he later extended for the Sasakian manifolds as well. In this present paper, we generalize the same characterization in generalized $g.f.f-$manifolds.

Criterions for constancy of the holomorphic sectional curvature and the antiholomorphic sectional curvature are proved for almost Hermitian manifolds. It is shown, that an almost Hermitian manifold satisfying the axiom of antiholomorphic planes or the axiom of antiholomorphic spheres is a real or a complex space form.

Local constancy of index for certain gradient mappings proved.

Properties of general Legendrian cycles $T$ acting in ${\mathbb R}^d\times S^{d-1}$ are studied. In particular, we give short proofs for certain uniqueness theorems with respect to the projections on the first and second component of such currents: In general, $T$ is determined by its restriction to the Gauss curvature…

Timelike Thomsen surfaces are timelike minimal surfaces that are also affine minimal. In this paper, we make use of both the Lorentz conformal coordinates and the null coordinates, and their respective representation theorems of timelike minimal surfaces, to obtain a complete global classification of these surfaces and…

It is well-known that for a surface in a 3-dimensional real space form the constancy of the mean curvature is equivalent to the harmonicity of the Gauss map. However, this is not true in general for surfaces in an arbitrary 3-dimensional ambient space. In this paper we study this problem for surfaces in an important an…

We studied the axiom of anti-invariant 2-spheres and the axiom of co-holomorphic $(2n+1)$-spheres. We proved that a nearly Kählerian manifold satisfying the axiom of anti-invariant 2-spheres is a space of constant holomorphic sectional curvature. We also showed that an almost Hermitian manifold $M$ of dimension $2m\geq…

COMBO network improves optical flow estimation by combining deep learning with brightness constancy.

In this paper, we introduce metallic maps between metallic Riemannian manifolds, provide an example and obtain certain conditions for such maps to be totally geodesic. We also give a sufficient condition for a map between metallic Riemannian manifolds to be harmonic map. Then we investigate the constancy of certain map…

Extends harmonic map theory to arbitrary surfaces.

Determinants remain constant along specific families of differential operators.

By applying an average method in PDE, we obtain a dichotomy between "constancy" and "infinity" of the warping functions on complete noncompact Riemannian manifolds for an appropriate isometric immersion of a multiply warped product manifold $N_1\times_{f_2} N_2 \times \cdots \times _{f_k} N_k\,$ into a Riemannian mani…

Here, a Finsler manifold (M, F) is considered with corresponding curvature tensor, regarded as 2-forms on the bundle of non-zero tangent vectors. Certain subspaces of the tangent spaces of M determined by the curvature are introduced and called k-nullity foliations of the curvature operator. It is shown that if the dim…

The study examines spectral rigidity in Ricci solitons and Einstein-type manifolds.

We show that, the solutions of the isoperimetric problem for small volumes are $C^{2,α}$-close to small spheres. On the way, we define a class of submanifolds called pseudo balls, defined by an equation weaker than constancy of mean curvature. We show that in a neighborhood of each point of a compact riemannian manifol…

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 …

We study some qualitative properties of ancient solutions of superlinear heat equations on a Riemannian manifold, with particular interest in positivity and constancy in space.

This paper explores constantly curved holomorphic 2-spheres in complex Grassmannian and confirms their rarity.

Proposes a new metric space example showing non-constant topological dimension.

We consider the evolution of the asymptotically hyperbolic mass under the curvature-normalized Ricci flow of asymptotically hyperbolic, conformally compactifiable manifolds. In contrast to asymptotically flat manifolds, for which ADM mass is constant during Ricci flow, we show that the mass of an asymptotically hyperbo…

Digital camera pipelines employ color constancy methods to estimate an unknown scene illuminant, in order to re-illuminate images as if they were acquired under an achromatic light source. Fully-supervised learning approaches exhibit state-of-the-art estimation accuracy with camera-specific labelled training imagery. R…

New surfaces in Lorentz-Minkowski space with constant mean curvature identified.

A curve is rectifying if it lies on a moving hyperplane orthogonal to its curvature vector. In this work, we extend the main result of [Chen 2017, Tamkang J. Math. 48, 209] to any space dimension: we prove that rectifying curves are geodesics on hypercones. We later use this association to characterize rectifying curve…

We prove that, if $Ω\subset \mathbb{R}^n$ is an open bounded starshaped domain of class $C^2$, the constancy over $\partial Ω$ of the function $\varphi(y) = \int_0^{λ(y)} \prod_{j=1}^{n-1}[1-t κ_j(y)]\, dt$ implies that $Ω$ is a ball. Here $k_j(y)$ and $λ(y)$ denote respectively the principal curvatures and the cut v…

The study examines null curves in specific geometric manifolds and their properties.

Wide neural networks become linear, with constant tangent kernel, due to Hessian scaling.

This paper investigates the relationship between algebraic quantum field theories and factorization algebras on globally hyperbolic Lorentzian manifolds. Functorial constructions that map between these two types of theories in both directions are developed under certain natural hypotheses, including suitable variants o…

We are concerned in this article with a classical question in spectral geometry dating back to McKean-Singer, Patodi and Tanno: whether or not the constancy of holomorphic sectional curvature of a complex $n$-dimensional compact Kähler manifold can be completely determined by the eigenvalues of its $p$-Laplacian for a …

Let $M$ be a Riemannian manifold and $Ω$ a compact domain of $M$ with smooth boundary. We study the solution of the heat equation on $Ω$ having constant unit initial conditions and Dirichlet boundary conditions. The purpose of this paper is to study the geometry of domains for which, at any fixed value of time, the nor…

We consider generic static spacetimes with Killing horizons and study properties of curvature tensors in the horizon limit. It is determined that the Weyl, Ricci, Riemann and Einstein tensors are algebraically special and mutually aligned on the horizon. It is also pointed out that results obtained in the tetrad adjust…

We show that a family of isolated complex hypersurface singularities with constant Milnor number may fail, in the strongest sense, to have constant bi-Lipschitz type. Our example is the Briac con--Speder family $X_t:=\{(x,y,z)\in\C^3 | x^5+z^{15}+y^7z+txy^6=0 \}$ of normal complex surface germs; we show the germ $(X_0,…

The paper defines and studies isoparametric submanifolds in Riemannian Hilbert manifolds.

In the previous work, the notion of the Finsleroid--Finsler space have been formulated and the necessary and sufficient conditions for the space to be of the Landsberg type have been found. In the present paper, starting with particular spray coefficients, we demonstrate how the Landsberg condition can explicitly appea…

Lie algebroid Yang-Mills theories are a generalization of Yang-Mills gauge theories, replacing the structural Lie algebra by a Lie algebroid E. In this note we relax the conditions on the fiber metric of E for gauge invariance of the action functional. Coupling to scalar fields requires possibly nonlinear representatio…

Neural networks with DAGs show linearity as width increases.

We study the dependence of geometric quantization of the standard symplectic torus on the choice of invariant polarization. Real and mixed polarizations are interpreted as degenerate complex structures. Using a weak version of the equations of covariant constancy, and the Weil-Brezin expansion to describe distributiona…

We prove an extension of Milnor-Wood inequalities to a geometric situation. We study representations of the fundamental group of a compact manifold into the isometry group of a product of rank one spaces of the same dimension and show an upper bound on the volume of the representation. When the target group is the isom…

V1 cortex reconstructs images as Poisson equation solutions with varying weights.

We describe co-adjoint orbits and Casimir functions for two-step free-nilpotent Lie algebras. The symplectic foliation consists of affine subspaces of the Lie coalgebra of different dimensions. Further, we consider left-invariant time-optimal problems on two-step Carnot groups, for which the set of admissible velocitie…

The paper proves rigidity theorems for forms on reductive symmetric spaces.

The paper characterizes integrability of tensors on manifolds.

Study canonical deformations of complex forms and their cohomology properties.

In the classical theory of toric manifolds polytopes appear in two guises -- as Newton polytopes of line bundles on the complex, and as moment polytopes on the symplectic side, the link between the two being established by the prequantizability condition on the cohomology class of the symplectic form. Here we give a co…