Study on flat connections with controlled irregularity.
problem Boundedness of algebraic flat connections with limited irregularity.
method Analysis of families of algebraic flat connections and holonomic D-modules.
result Established boundedness of families of algebraic flat connections with controlled irregularity.
Study of symplectically flat connections and their functionals on smooth manifolds.
problem Understanding symplectically flat connections and their functionals on smooth manifolds.
method Extend symplectically flat connections to ζ-flat connections, introduce functionals with zeroes as symplectically flat connections, study critical points of these functionals, describe characteristic classes of ζ-flat bundles. result Novel geometric flows and characteristic classes of ζ-flat bundles are described. Study information geometry of warped product spaces, finding special connections.
problem Understanding information geometry in warped product spaces.
method Examined warped products with dually flat connections, characterized connections on base space.
result Characterized connections on base space R>0 as α-connections with α=±1. Study projective connections on surfaces using osculating spaces.
problem Understanding projective connections on surfaces.
method Analyzing the second neighborhood and osculating behavior of integral curves.
result Geometrical interpretation of joint invariants of a related group.
The study examines connections and their curvatures on different types of bundles.
problem Understanding connections and curvatures on various bundle types.
method Analysis of connections and curvatures on fiber, principal, and vector smooth bundles.
result Investigations into the relationships between connections and curvatures on different bundle types.
Study on Einstein warped spaces with specific connections and curvature properties.
problem Characterizing Einstein warped spaces with quarter symmetric connections.
method Analyzing curvature, Ricci, and scalar tensors with quarter symmetric connections.
result Proves conditions under which an Einstein warped space becomes a Riemannian product space.
Study Hom-Lie algebroid connections on complex manifolds.
problem Irreducible connections on Hom-Lie algebroids.
method Proved moduli space structure using H-gauge theory.
result Moduli space has a Hausdorff Hilbert manifold structure.
Study non-integrable distributions with various affine connections.
problem Characterize non-integrable distributions in Riemannian manifolds with different connections.
method Obtain Gauss, Codazzi, and Ricci equations for non-integrable distributions with semi-symmetric metric, non-metric, and statistical connections.
result Find new examples of Einstein and distributions with constant scalar curvature.
Study flat connections on Courant algebroids using Lie groups.
problem Flatness conditions on Courant algebroids.
method Metric generalized connections, flatness condition analysis.
result Existence of compact simple Lie groups as building blocks for flat transitive Courant algebroids.
In this paper, we study the Einstein warped products and multiply warped products with a quarter-symmetric connection. We also study warped products and multiply warped products with a quarter-symmetric connection with constant scalar curvature. Then apply our results to generalized Robertson-Walker spacetimes with a q…
Study connections on Lie groupoids and stacks using Atiyah sequences.
problem No specific problem stated; general connections on Lie groupoids and stacks.
method Construct connections using Atiyah sequences associated with transversal tangential distributions.
result Detailed study and construction of connections on Lie groupoids and stacks.
Ehresmann connections in tangent categories
problem Generalizing Ehresmann connections to a categorical setting
method Introducing tangent categories and Ehresmann connections
result Defining and proving properties of Ehresmann connections in tangent categories
New connections defined for a specific geometric structure.
problem Characterizing manifolds with a paracontact structure.
method Introduced and studied a pair of associated Schouten-van Kampen affine connections.
result Curvature properties of the connections are obtained.
Defines semi-symmetric metric connections on differential forms.
problem Analyzing connections on differential forms.
method Defined and studied semi-symmetric metric connections, computed their curvature and Ricci tensors, and analyzed Lie derivatives.
result Derived Gauss-Codazzi-Ricci equations and properties of canonical, Schouten, and Vrancreanu connections.
Study of multiplicative connections in Lie groupoids.
problem Defining and understanding multiplicative connections in Lie groupoids.
method Definition and study of multiplicative connections satisfying compatibility with the groupoid structure.
result Identification of the obstruction to the existence of a multiplicative connection.
Motivated by the study of a certain family of classical geometric problems we investigate the existence of multiplicative connections on proper Lie groupoids. We show that one can always deform a given connection which is only approximately multiplicative into a genuinely multiplicative connection. The proof of this fa…
Study of minimal surfaces in 3D space with special connections.
problem Classification of minimal translation surfaces in Euclidean spaces with semi-symmetric connections.
method Analysis of singular minimal translation surfaces in a 3D Euclidean space with a semi-symmetric connection.
result Classification of singular minimal translation surfaces in Euclidean spaces with semi-symmetric connections.
Study of real and quaternionic Lie algebroid connections on manifolds.
problem Understanding moduli spaces of connections in real and quaternionic geometry.
method Proved the moduli space has a Hausdorff Hilbert manifold structure.
result Generalized results from complex vector bundles to real and quaternionic settings.
Holonomy group of Bismut connection on Vaisman manifolds is studied.
problem Analyzing the holonomy group of Bismut connection on Vaisman manifolds.
method Proved and computed the holonomy group for Vaisman manifolds, including solvmanifolds and Hopf manifolds.
result Holonomy group of Bismut connection on Vaisman manifolds is contained in U(n-1).
Study on HYM connections on Kähler manifolds, calculating moduli space virtual dimension.
problem Calculating the moduli space of Hermitian-Yang-Mills connections.
method Analytic proof of stability of Higgs bundles on compact Kähler manifolds.
result Virtual dimension of the moduli space of HYM connections calculated.
Deform moment map on symplectic connections using star product algebras.
problem Understanding symplectic connections and their deformations.
method Study vector bundle of Fedosov star product algebras, formal connection, curvature, and star product trace.
result Showed star product trace as a formal symplectic form and moment map.
The study connects conic connections and torsion-free principal connections on G-structures.
problem Relating torsion tensors of principal connections to characteristic conic connections.
method Formulating and verifying conditions for the existence of characteristic conic connections implying torsion-free principal connections.
result Conditions for the existence of characteristic conic connections imply the existence of torsion-free principal connections, verified for adjoint varieties of simple Lie algebras.
Study on connection points on double regular polygons, providing coordinates and proving non-connection points.
problem Identifying connection points on double regular polygons.
method Examined coordinates in trace field, provided constructive proof for prime n. result For n=7, conjectured all remaining points are connection points; for n≥7 prime, provided explicit separatrix. We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of (J2=±1)-metric manifolds. We characterize when such a connection is a…
This thesis studies moduli spaces of singular connections on 3-manifolds and manifolds with cylindrical ends. A Chern-Simons functional is defined for singular connections on 3-manifolds which are singular along a knot. The critical points of that Chern-Simons functional are flat singular connections. The Hodge-de Rham…
The paper studies geometric structures in Sol_3 with two connections.
problem Analyzing geometric structures in Sol_3 using specific connections.
method Used Levi-Civita and semi-symmetric non-metric connections to study Sol_3.
result Concluded geometric structures in Sol_3 with both connections.
Study jets of flat partial connections in foliations.
problem Characterize and understand flat partial connections in foliations.
method Define and apply jets to flat partial connections in smooth foliations and locally free sheaves, focusing on codimension one and arbitrary codimension foliations.
result Define and apply jets to characterize transversely affine and projective structures in foliations.
The paper studies para-Sasaki-like manifolds with a new metric connection.
problem Investigating new geometric structures on para-Sasaki-like manifolds.
method Deriving relations between connections, analyzing curvature tensors, studying solitons, constructing examples.
result Derived relations and properties of para-Sasaki-like manifolds with the generalized symmetric metric connection.
The paper studies a new connection on Riemannian manifolds and finds conditions for symplectic manifolds.
problem Exploring a new quarter-symmetric non-metric connection on Riemannian manifolds.
method Analyzes the properties and relations of the torsion tensor and curvature tensors of the new connection.
result Conditions for a manifold to be symplectic when endowed with the new connection.
The paper studies stability of F-Yang-Mills connections on complex projective spaces.
problem Stability of F-Yang-Mills connections on complex projective spaces.
method Inspired by Lawson-Simons, the paper proves stability conditions and structures for F-Yang-Mills connections.
result Conditions for weakly stable F-Yang-Mills connections on complex projective spaces.
We study a moduli space of ASD connections over S3×R. We consider not only finite energy ASD connections but also infinite energy ones. So the moduli space is infinite dimensional in general. We study the (local) mean dimension of this infinite dimensional moduli space. We show the upper bound on the …
Study higher genus polylogarithms under Riemann surface degenerations.
problem Understanding higher genus polylogarithms under degenerations.
method Investigate the Enriquez connection for polylogarithms and show it becomes a known connection for families of Riemann surfaces.
result Higher genus polylogarithms can be described explicitly as power series in deformation parameters and logarithms of families.
Study non-symmetric non-metric connections on Kenmotsu manifolds.
problem Properties of Kenmotsu manifolds with non-symmetric non-metric connections.
method Investigate curvature properties and irregularity of Kenmotsu manifolds.
result Kenmotsu manifolds with non-symmetric non-metric connections are irregular.
Study Yang-Mills connections on conformally compact manifolds, proving existence of extensions.
problem Existence of Yang-Mills connections on conformally compact manifolds.
method Proving existence of Yang-Mills connections for small deformations.
result Confirming the existence of Yang-Mills connections in the interior of the manifold.
Study lift metrics and connections on tangent bundles of Riemannian manifolds.
problem Investigate geometric properties of tangent bundles and their lifts.
method Analyze lift metrics and connections on TM of (M,g), and study statistical and Codazzi couples. result Prove a result on 1-Stein and Osserman structures on TM. Study of time-dependent metrics and connections in geometry.
problem Understanding geodesics and connections in time-dependent Riemannian manifolds.
method Examine connections on product manifolds, explore parallel transport, geodesics, and torsion.
result Define the derivative of a one-parameter family of connections.
Study Bismut connection curvatures and solve Yamabe and Calabi-Yau problems.
problem Yamabe problem and Calabi-Yau with torsion metrics for Bismut connection.
method Analysis of Bismut scalar and Ricci curvatures, construction of examples.
result Existence of metrics with constant Bismut scalar curvature.
We study Yang-Mills connections on holomorphic bundles over complex Kähler manifolds of arbitrary dimension, in the spirit of Hitchin's and Simpson's study of flat connections. The space of non-Hermitian Yang-Mills (NHYM) connections has dimension twice the space of Hermitian Yang-Mills connections, and is locally isom…
Study on solvable Lie groups with specific Weyl connections.
problem Characterizing solvable Lie groups with invariant stretched non-positive Weyl connections.
method Analyzing structure and classification of solvable Lie groups.
result Classification of solvable Lie groups and compact solvmanifolds with invariant SNP connections.
Study lower bounds for connectivity of distance function level sets in convex sets.
problem Understanding connectivity of distance function level sets in convex sets.
method Lower bound calculation using critical points of the distance function.
result Provide a lower bound for the range of connectivity.
Study flat GL(1|1) connections using fatgraphs and coordinates.
problem Understanding moduli space of flat GL(1|1) connections.
method Assigning coordinates to fatgraphs and using Whitehead moves for trivalent graphs.
result Closed explicit formula for Whitehead moves in trivalent graphs.
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.
The paper studies connectivity properties of Morse complexes as simplicial complexes grow.
problem Understanding connectivity of Morse complexes as simplicial complexes evolve.
method Bestvina-Brady Morse theory applied to a generalized Morse complex.
result Proves M(Δ) becomes arbitrarily highly connected as Δ grows. Study geodesics of meromorphic connections on Riemann surfaces.
problem Understanding the asymptotic behaviors of geodesics in meromorphic connections.
method Use branched affine structure induced by Fuchsian meromorphic connections.
result Examples of geodesics with infinitely many self-intersections and peculiar omega-limit sets.
Study Lie algebroid connections on principal bundles over complex projective varieties.
problem Existence and properties of Lie algebroid connections on principal bundles.
method Definition and study of Lie algebroid valued connections on holomorphic principal G-bundles, investigation of existence criteria.
result Investigation of criteria for existence of Lie algebroid connections on principal G-bundles over smooth complex projective curves.
Study analyzes error in ReLU networks with local connections.
problem Improving neural network performance and understanding approximation errors.
method Analyzed approximation error of ReLU networks with local connections.
result Error estimate depends on depth and width of hidden layers.
This article is a continuation of my former article "On Connectivity Spaces". After some brief historical references relating to the subject, separation spaces and then adjoint notions of connective representation and connective foliation are developed. The connectivity order previously defined only in the finite case …
Study investigates Hashiguchi connection's uniqueness and existence in Finsler geometry.
problem Investigating the Hashiguchi connection's uniqueness and existence in Finsler geometry.
method Intrinsic Finsler geometry using Klein-Grifone approach (KG-approach). Calculations for torsion and curvature tensors, examination of properties, and comparison of four fundamental linear connections.
result Investigates the Hashiguchi connection's uniqueness and existence in Finsler geometry.