arXiv research

Local generalization of frame bundles using a weakened Maurer-Cartan equation.

Generalizes abelianization for framed local systems over surfaces.

It is shown that the integrability conditions of the equations satisfied by the local Frenet frame associated with a holomorphic curve in a complex Grassmann manifold coincide with a special class of nonabelian Toda equations. A local moving frame of a holomorphic immersion of a Riemann surface into a complex Grassmann…

Given a Lie pseudo-group action, an equivariant moving frame exists in the neighborhood of a submanifold jet provided the action is free and regular. For local equivalence problems the freeness requirement cannot always be satisfied and in this paper we show that, with the appropriate modifications and assumptions, the…

Proposes SE(3) equivariant graph neural networks with local frames for efficient geometric approximation.

This article addresses the two significant aspects of Ozsváth and Szabó's knot Floer cube of resolutions that differentiate it from Khovanov and Rozansky's HOMFLY-PT chain complex: (1) the use of twisted coefficients and (2) the appearance of a mysterious non-local ideal. Our goal is to facilitate progress on Rasmussen…

Frame flows on certain symmetric spaces mix exponentially.

Clarifies Einstein-Cartan gravitation with Dirac spinor on generalized frame bundle.

An explicit model of fiber bundle with local fibers being disinct copies of vector 3-space is introduced. They are endowed with frames which are used as local isotopic ones. The field local of isotopic frames is considered as gauge field itself while the form of gauge connections is derived from it. A covariant equatio…

This paper is devoted to apply the equivariant moving frame method to study the local equivalence problem of third order ordinarily differential equation under the pseudo-group of fiber preserving transformations.

Study of lightcone framed surfaces in Lorentz-Minkowski 3-space, focusing on curvature behavior.

We synthesize and extend the previous ideas about appearance of both noncommutative and Finsler geometry in string theory with nonvanishing B--field and/or anholonomic (super) frame structures \cite{vstring,vstr2,vnonc,vncf}. There are investigated the limits to the Einstein gravity and string generalizations containin…

Proposes local coordinate frames for improving model performance in complex dynamical systems.

Paper proposes a CNN-based method for estimating intra frame bits and quality.

The paper contains a review on the general connection theory on differentiable fibre bundles. Particular attention is paid to (linear) connections on vector bundles. The (local) representations of connections in frames adapted to holonomic and arbitrary frames is considered.

New flow generates surfaces with constant curvature.

We generalize Ng's two-variable algebraic/combinatorial $0$-th framed knot contact homology for framed oriented knots in $S^3$ to knots in $S^1 \times S^2$, and prove that the resulting knot invariant is the same as the framed cord algebra of knots. Actually, our cord algebra has an extra variable, which potentially co…

Constructs Gabor frames for curved manifolds to detect boundaries.

The paper introduces a new concept of frame vorticity and uses it to find optimal sections in specific geometric settings.

Gradient descent constructs tight fusion frames.

An almost-Riemannian structure on a surface is a generalized Riemannian structure whose local orthonormal frames are given by Lie bracket generating pairs of vector fields that can become collinear. The distribution generated locally by orthonormal frames has maximal rank at almost every point of the surface, but in ge…

The paper studies conformal and projective structures using 2-frame bundles.

The article extends Thurston's Grafting Theorem to signed spaces and defines a framed monodromy map.

Localized Kasner-like singularities constructed in spacetime.

Lifts of maps to frame bundles studied for Riemannian manifolds.

A treatment in a neighborhood and at a point of the equivalence principle on the basis of derivations of the tensor algebra over a manifold is given. Necessary and sufficient conditions are given for the existence of local bases, called normal frames, in which the components of derivations vanish in a neighborhood or a…

Locally stable maps $S^3\to\mathbb{R}^4$ are classified up to homotopy through locally stable maps. The equivalence class of a map $f$ is determined by three invariants: the isotopy class $σ(f)$ of its framed singularity link, the generalized normal degree $ν(f)$, and the algebraic number of cusps $κ(f)$ of any extensi…

This paper proposes a representational model for image pairs such as consecutive video frames that are related by local pixel displacements, in the hope that the model may shed light on motion perception in primary visual cortex (V1). The model couples the following two components: (1) the vector representations of loc…

We exhibit a finite set of local moves that connect any two surgery presentations of the same 3-manifold via framed links in the three-sphere. The moves are handle-slides and blow-downs/ups of a particular simple kind.

We investigate the relationship between the algebra of tensor categories and the topology of framed 3-manifolds. On the one hand, tensor categories with certain algebraic properties determine topological invariants. We prove that fusion categories of nonzero global dimension are 3-dualizable, and therefore provide 3-di…

We pursue the analogy of a framed flow category with the flow data of a Morse function. In classical Morse theory, Morse functions can sometimes be locally altered and simplified by the Morse moves. These moves include the Whitney trick which removes two oppositely framed flowlines between critical points of adjacent i…

LLoCa makes any network Lorentz-equivariant, achieving high accuracy and efficiency.

Let $A\neq 0$ be a complex number with $|A|\neq 1$. Let $M$ be a compact smooth oriented $3$-manifold, the $SU(3)$-skein space of $M$, $S_A(M)$, is the vector space over $\mathbb{C}$ generated by framed oriented links (including framed oriented trivalent graphs in $M$) quotient by the $SU(3)$-skein relations due to Ku…

Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space Omega FTM on the bundle of frames in the tangent bundle of M. We show that Omega FTM is the group completion of LM, and p…

This paper proves exponential mixing for frame flows on hyperbolic manifolds with cusps.

A necessary condition for a connection in a vector bundle to be locally metric is for its curvature matrix, which consists of $2$ forms, to be skew symmetric with respect to some local frame. In this paper we give a simple algorithm that can be used to decide when a matrix of $2$ forms is equivalent to a skew symmetric…

We investigate the geometric properties of hyperbolic affine flat, affine minimal surfaces in the equiaffine space $\mathbb{A}^3$. We use Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of such surfaces and…

We investigate the geometric properties of lightlike surfaces in the Minkowski space $\R^{2,1}$, using Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of lightlike surfaces of constant type in $\R^{2,1}$ an…

The work extends the A. Connes' noncommutative geometry to spaces with generic local anisotropy. We apply the E. Cartan's anholonomic frame approach to geometrical models and physical theories and develop the nonlinear connection formalism for projective module spaces. Examples of noncommutative generation of anholonom…

If one could assume that local coordinates in a Riemannian manifold were orthogonal, then local expressions for differential operators, and curvature computations, would be simplified. It is always possible on 2-manifolds, using geometric normal coordinates or isothermal coordinates. In 1984, Dennis DeTurck and Dean Ya…

Single-molecule localization fluorescence microscopy constructs super-resolution images by sequential imaging and computational localization of sparsely activated fluorophores. Accurate and efficient fluorophore localization algorithms are key to the success of this computational microscopy method. We present a novel l…

Extends Ooguri-Vafa symplectic form to framed Higgs bundles.

Inspired by the observation that humans are able to process videos efficiently by only paying attention where and when it is needed, we propose an interpretable and easy plug-in spatial-temporal attention mechanism for video action recognition. For spatial attention, we learn a saliency mask to allow the model to focus…

In this paper we relate the geometric Poisson brackets on the Grassmannian of 2-planes in R^4 and on the (2,2) Moebius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Moebius sphere does not restrict to the space of differential invariants of Schwarzian type. But…

New theory defines invariants for framed 3-manifolds using trivalent graphs with self-loops.

Researchers found 5 local fields to uniquely describe 3D director fields, related through 6 differential relations.

Classifies gradient Ricci solitons with harmonic Weyl curvature in dimensions 5 and above.

The Ooguri-Vafa space is a 4-dimensional incomplete hyperkähler manifold, defined on the total space of a singular torus fibration with one singular nodal fiber. It has been proposed that the Ooguri-Vafa hyperkähler metric should be part of the local model of the hyperkähler metric of the Hitchin moduli spaces, near th…