arXiv research

We prove a conjecture of Gromov's to the effect that manifolds with isotropic curvature bounded below by 1 (after possibly rescaling) are macroscopically 1-dimensional on the scales greater than 1. As a consequence we prove that compact manifolds with positive isotropic curvature have virtually free fundamental groups.…

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

New method improves representation learning from non-i.i.d. and non-isotropic data.

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

Stochastic Gradient Langevin Dynamics infuses isotropic gradient noise to SGD to help navigate pathological curvature in the loss landscape for deep networks. Isotropic nature of the noise leads to poor scaling, and adaptive methods based on higher order curvature information such as Fisher Scoring have been proposed t…

The paper examines Randers metrics with isotropic scalar curvature properties.

Paper establishes a relation between Berwald scalar curvature and S-curvature.

In this paper, we find a condition on $(α, β)$-metrics under which the notions of isotropic S-curvature, weakly isotropic S-curvature and isotropic mean Berwald curvature are equivalent.

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

Study isotropic Riemannian maps and helices along them.

We study two types of isotropic planes: weakly isotropic and strongly isotropic planes. We prove that a Riemannian manifold of indefinite metric is conformally flat if and only if its curvature tensor vanishes on all the strongly isotropic planes. We specialize the plane axiom for Riemannian manifolds of indefinite met…

Study isotropic curves on complex quadric with geometric relations.

In this paper, we construct a new class of Finsler manifolds called generalized isotropic Berwald manifolds which is an extension of the class of isotropic Berwald manifolds. We prove that every generalized isotropic Berwald manifold is a generalized Douglas-Weyl manifold. On a compact generalized isotropic Berwald man…

Constructs a moment map flow for isotropic maps on surfaces.

Spinor representation in isotropic space via Laguerre geometry.

The paper studies hanging chains and surfaces in degenerate geometries.

Large-batch stochastic gradient descent (SGD) is widely used for training in distributed deep learning because of its training-time efficiency, however, extremely large-batch SGD leads to poor generalization and easily converges to sharp minima, which prevents naive large-scale data-parallel SGD (DP-SGD) from convergin…

Developed a new concept of isometric surfaces in isotropic space.

Paper shows isotropic $E$- and $S$-curvatures are equivalent in warped Finsler metrics.

In this paper we will show that a Lagrangian, Lorentzian surface $M^2_1$ in a complex pseudo space form $\widetilde M^2_1 (4c)$ is pseudo-isotropic if and only if $M$ is minimal. Next we will obtain a complete classification of all Lagrangian, Lorentzian surfaces which are lightlike pseudo-isotropic but not pseudo-isot…

The paper studies Kropina metrics with a specific curvature property.

New approach uses isotropic geometry to solve Euclidean problems.

We show that for $m>n\geq 2$, there are at least two exact isotropic $n$-tori in $\mathbb{C}^m$ which are not Hamiltonian isotopic in $\mathbb{C}^m$, even though they are smoothly isotopic as isotropic $n$-tori. We apply this discovery to obtain more distinct non-exact isotropic tori in $\mathbb{C}^m$.

Study classifies zero mean curvature surfaces with planar curvature lines.

The study of Laguerre isotropic hypersurfaces with rigidity and isoparametric properties.

This paper aims to provide a description of totally isotropic Willmore two-spheres and their adjoint transforms. We first recall the isotropic harmonic maps which are introduced by Hélein, Xia-Shen and Ma for the study of Willmore surfaces. Then we derive a description of the normalized potential (some Lie algebra valu…

Study surfaces with constant ratio of principal curvatures in Euclidean and isotropic geometries.

We classify translation surfaces in isotropic geometry with arbitrary constant isotropic Gaussian and mean curvature under the condition that at least one of translating curves lies in a plane.

This work extends holomorphic surface representations to isotropic space.

The study finds compact vacuum static spaces with positive isotropic curvature are spheres or products of a circle and sphere.

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 …

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation of time-like surfaces parametrized along isotropic directions in $R^{2,1}$, $R^{3,1}$ and $R^{2,2}$. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a gen…

In this work, we are interested in the differential geometry of surfaces in simply isotropic $\mathbb{I}^3$ and pseudo-isotropic $\mathbb{I}_{\mathrm{p}}^3$ spaces, which consists of the study of $\mathbb{R}^3$ equipped with a degenerate metric such as $\mathrm{d}s^2=\mathrm{d}x^2\pm\mathrm{d}y^2$. The investigation is…

The study examines surfaces in isotropic space with specific Gauss map properties.

The paper studies conformally flat cubic metrics with isotropic curvature, finding they must be Minkowski.

In this paper, we study the second approximate Matsumoto metric on a manifold M. We prove that F is of scalar flag curvature and isotropic S-curvature if and only if it is isotropic Berwald metric with almost isotropic flag curvature.

We consider smooth isotropic immersions from the 2-dimensional torus into $R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed Lagrangian torus of $R^4$. We prove that such isotropic immersions can be approximated by arbitrarily $C^0$-close piecewise linear isotropic maps. If $n \geq 3$ the piece…

In this work, we are interested in the differential geometry of curves in the simply isotropic and pseudo-isotropic 3-spaces, which are examples of Cayley-Klein geometries whose absolute figure is given by a plane at infinity and a degenerate quadric. Motivated by the success of rotation minimizing (RM) frames in Eucli…

Totally isotropic surfaces in $S^6$ are not necessarily Willmore surfaces. Therefore it is the first goal of this paper to derive a geometric characterization of totally isotropic Willmore two-spheres in $S^6$. This will naturally yield to a description of such surfaces in terms of the loop group language. Moreover, ap…

The paper bounds bandwidth and focal radius for manifolds with positive isotropic curvature.

This paper represents a preliminary (pre-reviewing) version of a sublinear variational algorithm for isotropic Gaussian mixture models (GMMs). Further developments of the algorithm for GMMs with diagonal covariance matrices (instead of isotropic clusters) and their corresponding benchmarking results have been published…

The isotropic 3-space I^3 which is one of the Cayley--Klein spaces is obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we give several classifications on the surfaces in I^3 with the constant relative curvature (analogue of the Gaussian cu…

In this paper, we study factorable surfaces in a 3-dimensional isotropic space. We classify such surfaces with constant isotropic Gaussian (K) and mean curvature (H). We provide a non-existence result related with the surfaces satisfying H/K=const. Several examples are also illustrated.

In this paper, the concept of isotropic projective Ricci curvature has been investigated. By classification of Randers metric of isotropic projective Ricci curvature, it is shown that Randers metric of projective Ricci curvature is reversible if and only if it is of square projective Ricci curvature.

The $\hat B_n^{(1)}$-hierarchy is constructed from the standard splitting of the affine Kac-Moody algebra $\hat B_n^{(1)}$, the Drinfeld-Sokolov $\hat B_n^{(1)}$-KdV hierarchy is obtained by pushing down the $\hat B_n^{(1)}$-flows along certain gauge orbit to a cross section of the gauge action. In this paper, we (1) u…

In this paper, we study the rotational surfaces in the isotropic 3-space I^3. satisfying Weingarten conditions in terms of the relative curvature K (analogue of the Gaussian curvature) and the isotropic mean curvature H. In particular, we classify such surfaces of linear Weingarten type in I^3.

New sprays of constant curvature introduced; conditions for metrizability given.

Classifies hypersurfaces with constant isotropic curvature in space forms.