New conformal invariants discovered in absolute parallelism geometry.
problem Investigating conformal changes in absolute parallelism geometry.
method Using the Weitzenböck and Levi-Civita connections of an absolute parallelism space.
result Found new conformal invariants.
A Lie group has a unique metric when viewed as a flat absolute parallelism.
problem Understanding different metrics on Lie groups and their curvature properties.
method Shifted perspective of viewing Lie groups as flat absolute parallelisms.
result Lie groups have a canonical metric when viewed in this new way.
Einstein's philosophy uses differential identities to derive GR field equations.
problem Constructing field theories in alternative geometries.
method Explains differential identities and their role in GR and PAP-geometry.
result Derived a more general differential identity in PAP-geometry.
The aim of the present paper is to construct and investigate a Finsler structure within the framework of a Generalized Absolute Parallelism space (GAP-space). The Finsler structure is obtained from the vector fields forming the parallelization of the GAP-space. The resulting space, which we refer to as a Finslerized Pa…
In this paper we provide a \emph{global} investigation of the geometry of parallelizable manifolds (or absolute parallelism geometry) frequently used for application. We discuss the different linear connections and curvature tensors from a global point of view. We give an existence and uniqueness theorem for a remarkab…
Study sub-Riemannian geometry on parallelizable distributions.
problem Investigate the geometry of sub-Riemannian structures on non-integrable parallelizable distributions.
method Construct and analyze two linear connections: Weitzenböck and sub-Riemannian.
result Two concrete examples (S^3 and S^7) illustrate the application of sub-Riemannian geometry.
Absolute parallelism geometry is frequently used for physical applications. It has two main defects, from the point of view of applications. The first is the identical vanishing of its curvature tensor. The second is that its autoparallel paths do not represent physical trajectories. The present work shows how these de…
In this paper, we study Absolute Parallelism (AP-) geometry on the tangent bundle TM of a manifold M. Accordingly, all geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. Moreover, many new geometric objects, which have n…
The paper classifies almost complex manifolds with non-degenerate torsion and their coverings.
problem Classification of almost complex manifolds with non-degenerate torsion.
method Analysis of homogeneous and non-homogeneous manifolds, classification of Lie algebras, covering spaces.
result Classification of manifolds with solvable Lie algebra up to coverings.
A field theory is constructed in the context of parameterized absolute parallelism geometry. The theory is shown to be a pure gravity one. It is capable of describing the gravitational field and a material distribution in terms of the geometric structure of the geometry used (the parallelization vector fields). Three t…
Develops an alternative approach to Tanaka's prolongation of geometric structures.
problem Generalizing Tanaka's prolongation of G-structures to a broader class of geometric structures.
method Alternative constructive approach based on quasi-gradations and filtered vector spaces.
result Constructs an alternative method for describing local invariants and automorphism groups of Tanaka structures.
Recently, it has been shown that Absolute Parallelism (AP) geometry admits paths that are naturally quantized. These paths have been used to describe the motion of spinning particles in a background gravitational field. In case of a weak static gravitational field limits, the paths are applied successfully to interpret…
We explicitly determine the structure equations of 5-dimensional Levi 2-nondegenerate CR hypersurfaces, using our recently constructed canonical Cartan connection for this class of CR manifolds. We also give an outline of the basic properties of absolute parallelisms and Cartan connections, together with a brief discus…
Develops a new method for constructing absolute parallelisms on CR structures.
problem Constructing absolute parallelisms for CR structures in arbitrary dimensions.
method Bigraded Tanaka prolongation procedure to construct canonical absolute parallelisms.
result Unique canonical absolute parallelism found for CR structures with maximal infinitesimal symmetry.
Affine λ-equidistants of convex polygons with parallel opposite sides have applications to isoperimetric inequalities.
problem Reconstruction and area estimates for affine λ-equidistants of convex polygons with parallel opposite sides. method Using Wigner caustics and centre symmetry sets.
result Proving a discrete version of the improved isoperimetric inequality.
We study grassmannians associated with a linear space with a nondegenerate hermitian form. The geometry of these grassmannians allows us to explain the relation between a (pseudo-)riemannian projective geometry and the conformal structure on its ideal boundary (absolute). Such relation encompasses, for instance, the us…
Generalized Tanaka prolongation ensures convergence of formal embeddings of complex manifolds.
problem Ensuring convergence of formal embeddings of complex manifolds under weaker conditions.
method Formulated and proved generalized Tanaka prolongation for geometric structures.
result Convergence of formal embeddings holds under weaker semi-positive normal bundle conditions.
We reduce to various absolute parallelisms, namely to certain {e}-structures on manifolds of dimensions 7, 6, 5, the biholomorphic equivalence problem or the intrinsic CR equivalence problem for generic submanifolds M^5 in C^4 of CR dimension 1 and of codimension 3 that are maximally minimal and are geometry-preserving…
In this paper we discuss curvature tensors in the context of Absolute Parallelism geometry. Different curvature tensors are expressed in a compact form in terms of the torsion tensor of the canonical connection. Using the Bianchi identities some other identities are derived from the expressions obtained. These identiti…
The paper studies geometries with parallel skew-symmetric torsion and their submersions.
problem Understanding geometries with parallel skew-symmetric torsion.
method Analyzing metric connections and submersions.
result Complete local classification of geometries with parallel skew-symmetric torsion in principal bundle cases.
Unified view of geometries with parallel skew torsion via submersions.
problem No de Rham decomposition for geometries with torsion.
method Developed and unified submersion constructions for geometries with parallel skew torsion.
result Completed and extended classification of irreducible geometries with parallel skew torsion.
Study left invariant spray geometry on Lie groups using parallel translations.
problem Understanding parallel translations in left invariant spray geometry.
method Using invariant frames and differential equations on Lie algebra, study parallel translations and curvature.
result Alternative interpretations and proofs of homogeneous curvature formulae.
Paper proposes a novel method to improve matrix completion with median loss for large datasets.
problem Matrix completion with absolute deviation loss for large-scale data.
method Proposes a refinement step using pseudo data to improve inefficient estimators of median matrix completion.
result Turns inefficient estimators into a rate (near-)optimal matrix completion procedure.
Inverts operators near parallel Ricci metric on non-compact manifolds.
problem Inverting curvature operators near a parallel Ricci metric.
method Local invertibility of operators in Sobolev spaces.
result Operators are locally invertible near a parallel Ricci metric.
In this paper, we deal with a generalization of the geometry of parallelizable manifolds, or the absolute parallelism (AP-) geometry, in the context of generalized Lagrange spaces. All geometric objects defined in this geometry are not only functions of the positional argument x, but also depend on the directional ar…
Extends square root velocity transform to handle curves in shape spaces of manifold-valued data.
problem Handling curves that leave the shape space in shape analysis.
method Generalizes absolutely continuous curves to strong Riemannian manifolds and extends SRVT to handle these curves.
result Computations in the SRVT framework can now be applied to curves in shape spaces of manifold-valued data.
The absolute Galois group of 3-manifolds determines their structure up to homeomorphism.
problem Determining the structure of 3-manifolds using their absolute Galois groups.
method Defined a relative absolute Galois group for 3-manifolds and used Chebotarev density properties and Hilbert ramification theory.
result Two branched covers of the three-sphere over a stably Chebotarev link are homeomorphic if and only if their absolute Galois groups are isomorphic.
Optimized parallel algorithms for identifying strong ties in data.
problem Identifying strong ties in data with varying distances and community sizes.
method Design and analysis of sequential and parallel algorithms for partitioned local depths.
result Optimized algorithms achieve up to 19.4x speedup in parallel execution.
Study of hypersurfaces in Sol4_0 geometry, classifying parallel and totally umbilical types.
problem Classifying hypersurfaces in the Sol4_0 geometry.
method Analyzing hypersurfaces with Codazzi tensors and parallel second fundamental forms.
result Full classification of hypersurfaces in Sol4_0, including parallel and totally umbilical types.
An immersion of a compact manifold is tight if it admits the minimal total absolute curvature over all immersions of the manifold. A prominent result in the study of minimal total absolute curvature immersions is the theorem of Chern and Lashof, which characterizes minimal total absolute curvature immersions, and tight…
Develops new approach to recover CR structures from their Levi foliations.
problem Recovering CR structures from their Levi foliations for nonregular symbols.
method Reduction to dynamical Legendrian contact structure on leaf space.
result New geometric interpretation of CR prolongation conditions.
Tractor calculus theory for curves in conformal geometry.
problem Understanding curves in conformal geometry.
method Tractor calculus for generic and null curves.
result Definition and expression of absolute conformal invariants.
We consider weighted parallel spinors in Lorentzian Weyl geometry in arbitrary dimensions, choosing the weight such that the integrability condition for the existence of such a spinor, implies the geometry to be Einstein-Weyl. We then use techniques developed for the classification of supersymmetric solutions to superg…
We classify non-dilatonic NS-NS type II supergravity backgrounds admitting a consistent absolute parallelism. They are all given by parallelised Lie groups admitting scalar flat bi-invariant lorentzian metrics. There are seven different classes, some of them containing moduli. For each class we determine the amount of …
eDCF estimates intrinsic dimension using local connectivity.
problem Challenges in estimating intrinsic dimension due to scale dependence.
method eDCF: a novel, scalable, and parallelizable method based on Connectivity Factor (CF).
result eDCF consistently matches leading estimators with comparable MAE and higher exact intrinsic dimension match rates.
Arithmetic Kleinian groups are distinguished by their finite quotients.
problem Distinguishing arithmetic Kleinian groups among all finitely generated residually finite groups.
method Constructing specific examples of arithmetic Kleinian groups and proving their profinite rigidity.
result Arithmetic Kleinian groups are uniquely identified by their finite quotients.
Study of symmetric affine surfaces with non-vanishing torsion.
problem Characterizing geometries of symmetric affine surfaces with torsion.
method Complete classification of local geometries assuming parallel torsion and local homogeneity for non-parallel torsion.
result Classification of local geometries for symmetric affine surfaces with torsion, generalizing previous results.
Study shape operator of relatively parallel hypersurfaces in n-dimensional geometry.
problem Characterize geometric properties of hypersurfaces in relative differential geometry.
method Analyze shape operator, principal curvatures, mean curvature, and affine normalization.
result Developed methods to study geometric properties of relatively parallel hypersurfaces.
Explains curves and surfaces in differential geometry.
problem Understanding smooth curves and surfaces in differential geometry.
method Problem-centered, elementary, visual approach focusing on essential techniques.
result Provides a solid foundation for further study in differential geometry.
This paper develops GPCA for probability distributions using Otto-Wasserstein geometry.
problem Analyzing modes of variation in datasets of probability measures.
method Geodesic Principal Component Analysis (GPCA) on Wasserstein space with neural networks.
result Identification of geodesic curves that capture modes of variation in probability distributions.
Study parallel tractors and cotractors on almost Grassmannian structures.
problem Characterize parallel tractors and cotractors on almost Grassmannian structures.
method Provide explicit formulae for splitting operators, first BGG operators, and prolongation connections. Characterize solutions of the BGG operators geometrically.
result Describe the geometry of the zero locus of solutions of the first BGG operators.
Study on charged parallel spinors and mass-charge inequalities.
problem Equality case of the spin positive mass theorem with charge.
method Investigation of charged parallel spinors and application to extremal charged manifolds.
result Characterization of the equality case of the mass-charge inequality.
Investigates parallel spinors on Lorentzian four-manifolds using differential geometry.
problem Characterizing and classifying Lorentzian four-manifolds with parallel spinors.
method Formulated parallel spinor flow equations and used parabolic pairs theory.
result Characterized all parallel Cauchy pairs on simply connected Cauchy surfaces and classified compact three-manifolds.
Introduces a natural parallel translation for navigation data.
problem Navigation data geometric representation and parallelism.
method Introduces a natural parallel translation using Riemannian parallelism.
result The natural parallel translation preserves the Randers norm and has a finite-dimensional holonomy group.
New geometric system from Hessian operators offers solutions to geometric problems.
problem Solving geometric problems using Hessian operators.
method Introducing a new differential-geometric system based on m-Hessian operators. result Deduced an a priori C1-estimate for solutions to the Dirichlet problem for m-Hessian equations. These notes introduce key techniques in differential geometry for curves and surfaces.
problem Understanding the basics of differential geometry for curve and surface analysis.
method Problem-centered, elementary, visual approach to teaching essential techniques.
result Provides a solid foundation for further study in differential geometry.
Unified description of string and brane worldvolumes using auto-parallel vector fields.
problem Describing worldvolumes of strings and branes in arbitrary backgrounds.
method Introducing auto-parallel generalised vector fields and their properties.
result Unified worldvolume equations for strings and branes.
I discuss geometry and normal forms for pseudo-Riemannian metrics with parallel spinor fields in some interesting dimensions. I also discuss the interaction of these conditions for parallel spinor fields with the condition that the Ricci tensor vanish (which, for pseudo-Riemannian manifolds, is not an automatic consequ…