arXiv research

Unified Schwarz lemma in Kähler and Hermitian geometry.

Survey of geometric flows from unified string theories.

Unified view of geometries with parallel skew torsion via submersions.

Unified study of Riemannian and sub-Riemannian geometries with synthetic Ricci curvature bounds.

Simplified proof of mass center system's uniqueness and generalized Pappus' theorem across Euclidean, spherical, and hyperbolic geometries.

Unified framework for exceptional and generalised geometry, and Poisson-Lie duality.

A new connection in Finsler geometry unifies various types of connections.

Unified geometry for relativity and beyond.

Unified description of aesthetic curves through self-affinities.

Unified framework for various geometric constructions.

Unified framework for continuous-state discrete flow matching models.

Unified description of string and brane worldvolumes using auto-parallel vector fields.

We use grid diagrams to present a unified picture of braids, Legendrian knots, and transverse knots.

Unified framework for complex, split-complex, and dual numbers.

A theorem of Lawson and Simons states that the only stable minimal submanifolds in complex projective spaces are complex submanifolds. We generalize their result to the cases of quaternionic and octonionic projective spaces. Our approach gives a unified viewpoint towards conformal and projective geometries.

In (Phys. Rev. D 62, 081501, 2000) we proposed a unified approach to description of continuous and discrete spacetime based on nonassociative geometry and described nonassociative smooth and discrete de Sitter models. In our paper we give the description of nonassociative Friedmann-Robertson-Walker spacetime.

Unified Jacobi coupling construction for various geometric settings.

Unified treatment of stability problems in geometry and analysis.

Unified framework for observables in n-plectic geometry.

Unified theory of measure-preserving diffusions on manifolds.

Unified approach to constructing integrable systems using Stäckel lifts.

Unified approach to invariants in equivariant geometry.

3-quasi-Sasakian manifolds were recently studied by the authors as a suitable setting unifying 3-Sasakian and 3-cosymplectic geometries. In this paper some geometric properties of this class of almost 3-contact metric manifolds are briefly reviewed, with an emphasis on those more related to physical applications.

In the framework of nonassociative geometry (hep-th/0003238) a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called "diodular discrete structure" which at large spacetime scales `looks like' a differentiable manifold. After a…

On a 6-dimensional real vector space $V$ there are three types of multisymplectic 3-forms. We present in this paper a unified treatment of these three types. Forms of each type represent a subset of $Λ^3 V^*$. In two cases they are open subsets, in the third one it is a submanifold of codimension 1. We study the geomet…

The paper studies Einstein-type manifolds with structural conditions.

We propose a unified approach to the theory of connections in the geometry of sprays and Finsler metrics which, in particular, gives a simple explanation of the well-known fact that all the classical Finslerian connections provide exactly the same formulas appearing in the calculus of variations.

Unified framework for Riemannian deep learning across manifold-valued representations.

Unified framework for Riemannian deep learning across manifold-valued representations.

Unified proof of end-point estimates for Radon transform on curved spaces.

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invar…

Unified framework for human motion generation on Riemannian manifolds.

Encoder-decoder networks using convolutional neural network (CNN) architecture have been extensively used in deep learning literatures thanks to its excellent performance for various inverse problems. However, it is still difficult to obtain coherent geometric view why such an architecture gives the desired performance…

Unified view of surfaces in R^n using Gauss map, caustics, and quadratic forms.

Geometric approach links hydrodynamic integrability to compatible nets.

We connect the algebraic geometry and representation theory associated to Freudenthal's magic square. We give unified geometric descriptions of several classes of orbit closures, describing their hyperplane sections and desingularizations, and interpreting them in terms of composition algebras. In particular, we show h…

$G$-deformability of maps into projective space is characterised by the existence of certain Lie algebra valued 1-forms. This characterisation gives a unified way to obtain well known results regarding deformability in different geometries.

Unified construction of compactifications using Grassmannian geometry.

Unified framework for Riemannian, Kahler, and hyper-Kahler geometries in 4D.

We revisit McLean's second variation formulas for calibrated submanifolds in exceptional geometries, and correct his formulas concerning associative submanifolds and Cayley submanifolds, using a unified treatment based on the (relative) calibration method and Harvey-Lawson's identities.

Unified framework for higher-order network analysis.

In this paper, it is elaborated the theory the Ricci flows for manifolds enabled with nonintegrable (nonholonomic) distributions defining nonlinear connection structures. Such manifolds provide a unified geometric arena for nonholonomic Riemannian spaces, Lagrange mechanics, Finsler geometry, and various models of grav…

A topology on a set $X$ is the same as a projection (i.e. an idempotent linear operator) $cl:2^X\to 2^X$ satisfying $A\subset cl(A)$ for all $A\subset X$. That's a good way to summarize Kuratowski's closure operator. Basic geometry on a set $X$ is a dot product $\cdot:2^X\times 2^X\to 2^Y$. Its equivalent form is an or…

The theory of $G$-structures provides us with a unified framework for a large class of geometric structures, including symplectic, complex and Riemannian structures, as well as foliations and many others. Surprisingly, contact geometry - the "odd-dimensional counterpart" of symplectic geometry - does not fit naturally …

We construct the space of infinitesimal variations for the Strominger system and an obstruction space to integrability, using elliptic operator theory. We initiate the study of the geometry of the moduli space, describing the infinitesimal structure of a natural foliation on this space. The associated leaves are relate…

Studies geometric mechanics for autonomous and nonautonomous systems.

We describe how to use the perturbation theory of Caffarelli to prove Evans-Krylov type $C^{2,α}$ estimates for solutions of nonlinear elliptic equations in complex geometry, assuming a bound on the Laplacian of the solution. Our results can be used to replace the various Evans-Krylov type arguments in the complex geom…

We investigate Nijenhuis deformations of $L_\infty$-algebras, a notion that unifies several Nijenhuis deformations, namely those of Lie algebras, Lie algebroids, Poisson structures and Courant structures. Additional examples, linked to Lie $n$-algebras and $n$-plectic manifolds, are included.