arXiv research

Study closed G2-structures with T3-symmetry, classifying them into types and deriving hypersymplectic structures.

Study of homogeneous spacetimes with isotropic symmetry and their symmetries.

Classifies vector equations with higher symmetries.

A proposal is made for what could well be the most natural symmetrical Riemannian spaces which are homogeneous but not isotropic, i.e. of what could well be the most natural class of symmetrical spaces beyond the spaces of constant Riemannian curvature, that is, beyond the spaces which are homogeneous and isotropic, or…

Generalizes kinematical Lie algebras for isotropic spacetimes.

Study isotropic embeddings of Lie group orbits and their application to magnetic geodesic flows.

Published in 1999, Christodoulou proved that the naked singularities of a self-gravitating scalar field are not stable in spherical symmetry and therefore the cosmic censorship conjecture is true in this context. The original proof is by contradiction and sharp estimates are obtained strictly depending on spherical sym…

New definition of Bäcklund transformation for surface isometric deformation.

Let (M,g) be a steady gradient Ricci soliton of dimension n \geq 4 which has positive sectional curvature and is asymptotically cylindrical. Under these assumptions, we show that (M,g) is rotationally symmetric. In particular, our result applies to steady gradient Ricci solitons in dimension 4 which are κ-noncollapsed …

Least symmetry breaking principle explains SGD's local minima in shallow ReLU networks.

In this paper we provide a systematic treatment of Willmore surfaces with orientation reversing symmetries and illustrate the theory by (old and new) examples. We apply our theory to isotropic Willmore two-spheres in $S^4$ and derive a necessary condition for such ( possibly branched) isotropic surfaces to descend to (…

The paper classifies spherically symmetric sprays and their curvature properties.

Researchers generalize cosmological models using Finsler geometry.

The isotropic correlation model explains equity returns better than linear factor models.

By adapting a technique of Molchanov, we obtain the heat kernel asymptotics at the sub-Riemannian cut locus, when the cut points are reached by an $r$-dimensional parametric family of optimal geodesics. We apply these results to the bi-Heisenberg group, that is, a nilpotent left-invariant sub-Rieman\-nian structure on …

Proposes Gaussian process priors on graph sets with geometric structure.

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

Posing Kepler's problem of motion around a fixed "sun" requires the geometric mechanician to choose a metric and a Laplacian. The metric provides the kinetic energy. The fundamental solution to the Laplacian (with delta source at the "sun") provides the potential energy. Posing Kepler's three laws (with input from Gali…

The paper examines Randers metrics with isotropic scalar curvature properties.

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

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

New quasi space forms solve Thurston's geometrical space form problem.

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.

The study finds conditions for flat isotropic non-homogeneous tori to be Hamiltonian minimal.

Spinor representation in isotropic space via Laguerre geometry.

The paper studies hanging chains and surfaces in degenerate geometries.

We create real-time geodesic rendering for non-isotropic geometries.

Study on generalized $m$-Kropina metrics in modified gravity and cosmology.

The thesis explores kinematical symmetries beyond Lorentzian spacetime.

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…

This work extends holomorphic surface representations to isotropic space.

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.

Stability of Schwarzschild singularity in near-Schwarzschild black holes under perturbations.