Discussing moving frames for curve and surface invariants.
problem Identifying differential invariants of curves and surfaces.
method Using moving frames for Euclidean, affine, and conformal transformations.
result Determine differential invariants of curves and surfaces.
New method for calculating Finsler connections using moving frame formalism.
problem Calculating Finsler connections for metrics depending on moving frame.
method Developed a moving frame formalism to simplify Finsler connection calculations.
result Derived formulae for Finsler connections when metric depends on moving frame.
New method studies moving points on curves using rotating frames.
problem Understanding the motion of points on curves.
method Constructing rotating frames for curves and analyzing the motion of points within these frames.
result A new binary mathematical formation mechanism for curves based on linear and rotational motion.
New proof of Lie-Tresse theorem with computational advantages.
problem Equivalence problem for PDEs under point transformations.
method Involutive moving frames and constructive moving frame method.
result First general upper bound on minimal differential invariants.
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…
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.
Develops algorithms for constructing moving frames and differential invariants for Lie group actions.
problem Equivalence of submanifolds under Lie group actions.
method Introduces two algorithms: a product group algorithm and a recursive algorithm.
result Constructs differential invariants and moving frames for groups of transformations on the plane.
Develops a new method for solving equivalence problems in pseudo-groups.
problem Solving equivalence problems in pseudo-groups.
method Combining Cartan's equivalence method and equivariant moving frame for pseudo-groups.
result A hybrid equivalence method that extends and illuminates its two progenitors.
New framed moves extend classical knot theory results.
problem Extending classical knot theory to framed braids.
method Introduced framed versions of L-moves, Hilden, Pure Hilden groups, and framed versions of the Birman theorem.
result Proved a framed version of the Birman theorem for framed links in plat representation.
New method for curve comparison using iterated integrals and moving frames.
problem Comparing curves robustly to noise and transformations.
method Moving frame method paired with log-signature transform.
result Algorithmic construction of invariants for curve equivalence under rigid motions.
In this paper we study the general affine geometry of curves in affine space A2. For a regular plane curves we define two kinds of moving frames. The first is of minimal order in all moving frames.The second is the Frenet moving frame. We get the moving equations of these moving frames. And we prove that curvature a…
Analyses cohomology relations for moving frames and coframes.
problem Relating Hopf cyclic cohomology of moving frames and coframes.
method Uses van Est analogy for DG Hopf algebras.
result Establishes cohomology isomorphism for DG Hopf algebras.
Minimal sets of moves for rotational Reidemeister diagrams are identified.
problem Understanding the minimal sets of moves for rotational Reidemeister diagrams.
method Detailed description and proof of minimal generating sets for rotational Reidemeister moves.
result Minimal generating sets for oriented, framed links contain 5 moves.
New formula connects Loewner energy to moving frames' renormalised energy.
problem Calculating Loewner energy of Jordan curves.
method Using renormalised energy of moving frames.
result Loewner energy as Kähler potential for Weil-Petersson space.
Parseval frames can be thought of as redundant or linearly dependent coordinate systems for Hilbert spaces, and have important applications in such areas as signal processing, data compression, and sampling theory. We extend the notion of a Parseval frame for a fixed Hilbert space to that of a moving Parseval frame for…
New neural network processes 3D volumes with improved equivariance.
problem Improving neural network performance on 3D volumes with symmetries.
method Equivariant neural network using moving frames approach.
result Trained model outperforms benchmarks in medical volume classification.
New SCI index bounds unknotting framed unknots.
problem Bounding unknotting framed unknots.
method Introducing Self-Crossing Index (SCI) and using it to provide bounds.
result SCI provides bounds for unknotting framed unknots.
Bertrand framed surfaces defined in Euclidean 3-space with applications.
problem Defining and characterizing Bertrand framed surfaces.
method Using moving frames to define Bertrand framed surfaces and analyzing their caustics and involutes.
result Conditions for caustics and involutes to be inverse operations of framed surfaces.
Framed flow categories were introduced by Cohen-Jones-Segal as a way of encoding the flow data associated to a Floer functional. A framed flow category gives rise to a CW-complex with one cell for each object of the category. The idea is that the Floer invariant should take the form of the stable homotopy type of the r…
Frames for Rn can be thought of as redundant or linearly dependent coordinate systems, and have important applications in such areas as signal processing, data compression, and sampling theory. The word "frame" has a different meaning in the context of differential geometry and topology. A moving frame for the tang…
Normal forms and invariants for nondegenerate hypersurfaces in C^2.
problem Equivalence problem for nondegenerate real hypersurfaces in C^2.
method Equivariant moving frames and invariant differentiation.
result A single real differential invariant of order 7 generates the entire algebra of differential invariants for nondegenerate real hypersurfaces at singularly umbilic points.
A theorem of Kirby gives a necessary and sufficient condition for two framed links in S^3 to yield orientation-preserving diffeomorphic results of surgery. Kirby's theorem is an important method for constructing invariants of 3-manifolds. In this paper, we prove a variant of Kirby's theorem for null-homologous framed l…
Classifies CR-embedded 3-folds in 5D hyperquadrics using moving frames.
problem Classifying CR-embedded 3-folds in 5D hyperquadrics.
method Applying Cartan's moving frames method to classify CR-embedded manifolds.
result Identifies homogeneous CR 3-folds in 5D hyperquadrics.
The paper uses Cartan moving frames to analyze data manifolds and neural network outputs.
problem Understanding the geometry and explainability of neural network outputs.
method Employing Cartan moving frames to study the Riemannian structure of data manifolds and their curvature.
result The relationship between neural network outputs and the geometry of inputs is exploited for explainable AI.
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…
The paper studies Riemannian four-manifolds and their twistor spaces using a moving frame approach.
problem Understanding the twistor spaces of Riemannian four-manifolds.
method Using the moving frame approach to analyze the twistor space Z of an oriented Riemannian four-manifold M. result Proves that first-order linear conditions on the almost complex structures of Z force the manifold M to be self-dual, and shows that the Atiyah-Hitchin-Singer twistor space bears a resemblance to a nearly Kähler manifold under first-order quadratic conditions. We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is prove…
The method of moving frames in Lie sphere geometry has produced significant results in the classification of Dupin hypersurfaces in spheres. What is the secret of its effectiveness? The answer emerges in the classification of nonumbilic isoparametric surfaces in the space form geometries. Using the method of moving fra…
New method constructs relative invariants for group actions on extended manifolds.
problem Constructing relative invariants for Lie group actions.
method Developed a constructive modification of the moving frame method for extended manifolds.
result Invariantization of the multiplier yields a canonical relative invariant of weight -1.
Study curvature of piecewise metrics using moving frames.
problem Deriving a curvature measure for piecewise-smooth Riemannian metrics.
method Used moving frame techniques to derive curvature, showing it satisfies Cartan structure equations and gauge transformation law.
result Equivalence of the derived curvature to existing densitized distributional curvature.
Study geometry of surfaces glued along a curve.
problem Geometry of surfaces glued along a curve.
method Moving frame along the curve, developable surfaces defined.
result Developed geometric properties of glued surfaces.
Diagonalizes metrics of 3D Lorentzian manifolds.
problem Diagonalizing metrics of 3D Lorentzian manifolds.
method Applying the technique of moving frames.
result Every smooth Lorentzian 3-manifold admits an atlas with a diagonal metric.
Noether's Theorem yields conservation laws for a Lagrangian with a variational symmetry group. The explicit formulae for the laws are well known and the symmetry group is known to act on the linear space generated by the conservation laws. The aim of this paper is to explain the mathematical structure of both the Euler…
The Frenet frame generalizes the Park transform for multi-phase circuits.
problem Generalizing the Park transform for multi-phase circuits.
method Using the Frenet frame and Cartan's moving frames.
result The Frenet frame provides a new approach to circuit analysis.
Generators and relations found for foam categories.
problem Understanding categories of framed tangled webs and foams.
method Multijet transversality techniques.
result Presentation by generators and relations of categories.
The paper is devoted to study the Dirichelet energy of moving frames on 2-dimensional tori immersed in the euclidean 3≤m-dimensional space. This functional, called Frame energy, is naturally linked to the Willmore energy of the immersion and on the conformal structure of the abstract underlying surface. As first …
SQAIR generates videos of moving objects, tracking and predicting them reliably.
problem Generating videos of moving objects with reliable object tracking and prediction.
method Explicitly encoding object presence, locations, and appearances in latent variables.
result SQAIR reliably discovers and tracks objects throughout sequences of frames and generates future frames.
A proof that hyperbolic plane cannot be immersed in Euclidean 3-space.
problem Proving the impossibility of isometrically immersing the hyperbolic plane in Euclidean 3-space.
method Applying ideas from undergraduate mathematics, including moving frames and connection forms, to simplify the proof.
result A key transition from principal directions to asymptotic directions simplifies the proof and yields a coordinate system.
Study on focal surfaces and evolutes of framed curves in hyperbolic 3-space using Legendrian duality.
problem Investigate differential geometry properties of framed curves, including singular points.
method Use Legendrian dualities to analyze focal surfaces and evolutes of hyperbolic framed curves.
result Show the relationship among focal surfaces, evolutes, and dual surfaces of evolutes.
A calculus modifies flow categories without changing their homotopy type.
problem Modifying flow categories without altering their homotopy type.
method A calculus of moves to modify framed flow categories.
result Two flow categories with stable homotopy type give move equivalent categories.
Study of curves in Lie sphere geometry using moving frames and variational principles.
problem Characterize curves in Lie sphere geometry using Lie curvatures.
method Moving frames, exterior differential systems, and calculus of variations.
result Critical curves are uniquely determined by Lie curvatures.
Kirby proved that two framed links in S^3 give orientation-preserving homeomorphic results of surgery if and only if these two links are related by a sequence of two kinds of moves called stabilizations and handle-slides. Fenn and Rourke gave a necessary and sufficient condition for two framed links in a closed, orient…
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.
The paper proves unique properties of Riemannian twistor spaces under specific curvature conditions.
problem Characterizing Riemannian twistor spaces under vanishing curvature conditions.
method Moving frame method, classification, and nonexistence proofs.
result The only twistor space with parallel Bochner tensor is CP3. Study on evolutes and focal surfaces of pseudo-spherical framed immersions in anti-de Sitter space.
problem Investigating singularities of evolutes and focal surfaces of pseudo-spherical framed immersions.
method Introduced pseudo-spherical non-null framed curves, defined moving frames, and analyzed evolutes and focal surfaces.
result Evolutes of pseudo-spherical framed immersions are the sets of singular points of their focal surfaces.
In this paper, we characterize Riemannian 4-manifold in terms of its almost Hermitian twistor spaces (Z,gt,J±). Some special metric conditions (including Balanced metric condition, first Gauduchon metric condition) on (Z,gt,J±) are studied. For the first Chern form of a natural unitary…
In recent works, the authors considered various Lagrangians, which are invariant under a Lie group action, in the case where the independent variables are themselves invariant. Using a moving frame for the Lie group action, they showed how to obtain the invariantized Euler-Lagrange equations and the space of conservati…
Study on minimal surfaces in Heisenberg group with duality formula.
problem Understanding minimal surfaces in Heisenberg group.
method Introducing transformation surfaces and using logarithmic derivative of moving frame.
result Derivation of Sym formula for dual minimal surface.