New connections found with specific torsion properties.
problem Understanding metric connections with specific torsion properties.
method Described Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
result Found new Lorentzian manifolds with metric connections having parallel, skew-symmetric torsion.
All radially symmetric connections on round spheres are parallel.
problem Classifying radially symmetric connections in vector bundles over spheres.
method Proving all such connections are parallel.
result Radially symmetric connections on round spheres are parallel.
Study on BAS manifolds with parallel torsion and curvature.
problem Characterizing and classifying BAS manifolds.
method Canonical reduction theorem, classification in homogeneous settings, construction of combined geometries.
result Classification of BAS manifolds in various settings.
Study on special Hermitian manifolds with specific connection properties.
problem Compact Hermitian manifolds with a particular Chern connection.
method Proved structure theorems for manifolds with parallel torsion and curvature.
result Structure theorems for manifolds with these properties.
Study on pseudo parallel contact CR-submanifolds in Kenmotsu manifolds.
problem Characterizing pseudo parallel contact CR-submanifolds in Kenmotsu manifolds.
method Analysis of pseudo parallel contact CR-submanifolds with respect to both Levi-Civita and semisymmetric metric connections in Kenmotsu manifolds.
result Equivalent conditions for pseudo parallel contact CR-submanifolds in Kenmotsu manifolds.
The axiomatic approach to parallel transport theory is partially discussed. Bijective correspondences between the sets of connections, (axiomatically defined) parallel transports, and transports along paths satisfying some additional conditions, are constructed. In particular, the equivalence between the concepts "conn…
A vector bundle with connection over a supermanifold leads naturally to a notion of parallel transport along superpaths. In this note we show that {\it every} such parallel transport along superpaths comes form a vector bundle with connection, at least when the base supermanifold is a manifold.
Classifies Riemannian manifolds with specific torsion properties.
problem Classifying Riemannian manifolds with parallel, non-twistorial torsion.
method Classifies complete simply connected Riemannian manifolds with a metric connection having parallel torsion, non-zero vectorial component, and zero twistorial component.
result Classifies complete simply connected Riemannian manifolds with the specified torsion properties.
Classifies simply-connected pluriclosed manifolds with parallel Bismut torsion.
problem Classifying specific types of manifolds with parallel Bismut torsion.
method Complete classification through mathematical analysis.
result Established a splitting theorem for certain manifolds.
Study invariant and anti-invariant submanifolds in special quasi-Sasakian manifolds.
problem Characterize invariant and anti-invariant submanifolds in special quasi-Sasakian manifolds.
method Investigate Chaki-pseudo parallel and Deszcz-pseudo parallel invariant submanifolds with Levi-Civita and semisymmetric metric connections. Also study invariant and anti-invariant submanifolds with Ricci solitons.
result Invariant and anti-invariant submanifolds are equivalent under certain conditions.
Anisotropic connections and parallel transport defined in Finsler spacetimes.
problem Defining and characterizing anisotropic connections and parallel transport in Finsler spacetimes.
method Introducing a new covariant derivative and parallel transport, identifying vertically trivial Finsler connections with anisotropic connections, and characterizing the Levi-Civita-Chern anisotropic connection.
result Characterization of the Levi-Civita-Chern anisotropic connection as the one preserving the length of parallely propagated vectors.
Classifies Calabi hypersurfaces with parallel Fubini-Pick form.
problem Classifying Calabi hypersurfaces with specific geometric properties.
method Classification based on parallel Fubini-Pick form and Levi-Civita connection.
result Classification of 2 and 3-dimensional Calabi hypersurfaces.
Study on instantons in G2 and Spin(7) manifolds.
problem Investigate instanton properties in G2 and Spin(7) manifolds. method Analyze the characteristic and torsion connections on G2 and Spin(7) manifolds, focusing on parallel torsion and Lee forms. result Conditions for the curvature of the characteristic connection to be a G2 instanton and for the torsion connection to be a Spin(7) instanton are established. 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.
The aim of the present paper is to investigate conformal changes in absolute parallelism geometry. We find out some new conformal invariants in terms of the Weitzenböck connection and the Levi-Civita connection of an absolute parallelism space.
The paper explores Lorentzian connections with parallel skew torsion.
problem Understanding metric connections with parallel skew-symmetric torsion in Lorentzian signature.
method Analyzing holonomy algebras, torsion, and curvature; constructing examples; classifying homogeneous spaces.
result Complete classification of Lorentzian naturally reductive homogeneous spaces in low dimensions.
In this short note we give an elementary proof of the fact that connections and their geometric parallel-transport counterpart are equivalent notions.
This paper surveys parallel submanifolds in Riemannian and pseudo-Riemannian manifolds.
problem Understanding parallel submanifolds in Riemannian and pseudo-Riemannian manifolds.
method Comprehensive survey of parallel submanifolds.
result Extrinsic invariants of parallel submanifolds do not vary from point to point.
A review of the parallel transport (translation) in fibre bundles is presented. The connections between transports along paths and parallel transports in fibre bundles are examined. It is proved that the latter ones are special cases of the former.
Let M be a simply connected Riemannian symmetric space, with at most one flat direction. We show that every Riemannian (or unitary) vector bundle with parallel curvature over M is an associated vector bundle of a canonical principal bundle, with the connection inherited from the principal bundle. The problem of finding…
Complete classification of centroaffine hypersurfaces with parallel cubic form.
problem Characterizing centroaffine hypersurfaces with specific geometric properties.
method Analyzing hypersurfaces with respect to the Levi-Civita connection of the centroaffine metric.
result A complete classification of locally strongly convex centroaffine hypersurfaces with parallel cubic form.
New connections found for quaternionic and para-quaternionic structures.
problem Characterizing integrability of generalized quaternionic and para-quaternionic structures.
method Defined and characterized integrability with respect to a abla-bracket on the generalized tangent bundle. result Existence of a canonical connection for these structures.
Characterizes Hermitian manifolds with Bismut parallel torsion.
problem Understanding curvature properties of Hermitian manifolds with specific torsion.
method Analyzes Bismut curvature tensor and parallel torsion conditions.
result Characterizes Bismut torsion parallel manifolds using Bismut curvature alone.
New method for non-abelian parallel transport in higher gauge theories.
problem Limitation of previous approaches to higher gauge fields, making them locally gauge equivalent to abelian connections.
method Generalizing the notion of higher connection to overcome fake flatness condition.
result Definition of a generalized higher holonomy functor free from fake flatness condition.
Explains linearizing a nonlinear connection on a pullback bundle.
problem Clarifying the geometric meaning of linearized connections.
method Fiberwise linear approximation of a vector bundle connection.
result Clarifies the geometric meaning of linearized connections.
As is well known, a metric on a manifold determines a unique symmetric connection for which the metric is parallel: the Levi-Civita connection. In this paper we investigate the inverse problem: to what extent is the metric of a Riemannian manifold determined by its Levi-Civita connection? It is shown that for a generic…
In this paper we introduce a notion of parallel transport for principal bundles with connections over differentiable stacks. We show that principal bundles with connections over stacks can be recovered from their parallel transport thereby extending the results of Barrett, Caetano and Picken, and Schreiber and Waldof f…
Study on null-projectability of Levi-Civita connections in neutral metrics.
problem Characterizing projectability of Levi-Civita connections along null parallel distributions.
method Analyzing projectability of torsion-free connections along foliations on manifolds, focusing on neutral metric signatures and mid-dimensional distributions.
result Extension of Patterson and Walker's Riemann extension metrics to null parallel distributions of any dimension.
New criterion for Ricci-flat manifolds with non-vanishing Rosenberg index.
problem Existence of parallel spinors on Ricci-flat manifolds.
method Generalization of existing results to manifolds with non-vanishing Rosenberg index.
result Every closed connected Ricci-flat spin manifold of dimension ≥ 2 with non-vanishing Rosenberg index has special holonomy.
Parallel spinors help characterize G2* structures and isotropic forms.
problem Characterizing G2* structures and isotropic forms on pseudo-Riemannian manifolds.
method Using a correspondence between irreducible parallel spinors and solutions of a differential system for three-forms.
result Explicit description of isotropic irreducible spinors in signature (4,3) and characterization of G2* structures.
Proof of smooth radial sections in vector bundles.
problem Existence of smooth radial sections in vector bundles.
method Proof of existence using linear connection.
result Existence of smooth radial sections in vector bundles.
Constructs parallel transport in higher gauge theory using principal 2-bundles.
problem Higher gauge theory and parallel transport in categorified principal bundles.
method Explicit construction of parallel transport for connections on principal 2-bundles.
result Proves constructions fit into axiomatic framework for categorified parallel transport.
We consider the more general question as to when a connection is a metric connection. There are two aspects to this investigation: first, the determination of the integrability conditions that ensure the existence of a local parallel metric in the neighbourhood of a given point and second, the characterization of the t…
Study of convex hypersurfaces with specific curvature properties.
problem Characterizing convex hypersurfaces with vanishing Weyl curvature and semi-parallel cubic form.
method Analyzing locally strongly convex affine hypersurfaces with vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of affine metric.
result Classification of such hypersurfaces, excluding flat affine metric cases.
The aim of this article is to study rational parallelisms of algebraic varieties by means of the transcendence of their symmetries. The nature of this transcendence is measured by a Galois group built from the Picard-Vessiot theory of principal connections.
Parallel transport of a connection in a smooth fibre bundle yields a functor from the path groupoid of the base manifold into a category that describes the fibres of the bundle. We characterize functors obtained like this by two notions we introduce: local trivializations and smooth descent data. This provides a way to…
Mathematical theory of super fiber bundles and connections developed.
problem Modeling anticommuting fermionic fields in mathematical physics.
method Detailed introduction to super fiber bundles, relative supermanifolds, and connections; construction of parallel transport map.
result Construction and comparison of parallel transport map with other methods in the literature.
This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold M. A superpath in M is, loosely speaking, a path in M together with an odd vector field in M along the path. We also develop a notion of parall…
Pole ladder improves parallel transport in affine spaces, showing exact results in symmetric spaces.
problem Improving numerical stability and accuracy in parallel transport algorithms.
method Developed a third-order parallel transport scheme using pole ladder in affine connection spaces, showing exact results in symmetric spaces.
result Pole ladder is a third-order scheme in general affine connection spaces and is exact in locally symmetric spaces.
The study characterizes Hermitian manifolds with parallel Bismut-Strominger torsion.
problem Characterizing Hermitian manifolds with specific torsion properties.
method Analyzing the curvature tensor and properties of the Bismut-Strominger connection.
result A necessary and sufficient condition for Bismut torsion parallel manifolds.
Let π:E→M be a vector bundle over a simply connected manifold and ∇ a linear connection in π. Let σ:U→E be a ∇-parallel section of π defined on a connected open subset U of M. We give sufficient conditions on U in order to extend σ to the whole M. We mainly concentrate to…
We classify locally homogeneous quasi-Sasakian manifolds in dimension five that admit a parallel spinor ψ of algebraic type F⋅ψ=0 with respect to the unique connection ∇ preserving the quasi-Sasakian structure and with totally skew-symmetric torsion. We introduce a certain conformal transformation of …
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.
The study explores properties of homologically locally connected spaces and their connections to other topological concepts.
problem Characterizing and understanding homologically locally connected spaces.
method Investigates properties and characterizations of homologically UVn-maps and lcGn-spaces, comparing them to existing concepts. result Identifies similarities and parallels between homologically locally connected spaces and other topological concepts.
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.
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.
There are several kinds of classification problems for real hypersurfaces in complex two-plane Grassmannians G2(Cm+2). Among them, Suh classified Hopf hypersurfaces M in G2(Cm+2) with Reeb parallel Ricci tensor in Levi-Civita connection. In this paper, we introduce a new notion of gene…
In this paper we investigate the relationship between the existence of parallel semi-Riemannian metrics of a connection and the reducibility of the associated holonomy group. The question as to whether the holonomy group necessarily reduces in the presence of a specified number of independent parallel semi-Riemannian m…