Explores connective spaces, their representations, foliations, and relations to diffeological spaces.
problem Developing a comprehensive theory of connective spaces and their properties.
method Historical context, development of connective representation and foliation, generalization of connectivity order, study of functorial relations with diffeological spaces.
result Connectivity order generalized to all connectivity spaces and connective foliations.
Study on submanifolds in generalized Sasakian-space-forms with various connections.
problem Analyzing submanifolds in generalized Sasakian-space-forms with different connections.
method Examines submanifolds in generalized Sasakian-space-forms with semisymmetric metric, non-metric, Schouten-van Kampen, and Tanaka-webster connections.
result Provides results on submanifolds in generalized Sasakian-space-forms with respect to various connections.
New C-connection characterizes 4D spaces conformal to Einstein spaces.
problem Characterizing 4D spaces conformal to Einstein spaces.
method Introducing C-connection, a Weyl connection that preserves conformal invariance. result Characterizes non-degenerate spaces conformal to Einstein spaces.
Researchers create earring spaces from metric spaces to study fundamental groups.
problem Understanding fundamental groups of constructed spaces from metric spaces.
method Attach copies of the circle to a dense subset of a separable metric space to form an earring space, then analyze its fundamental group.
result The fundamental group of the earring space is isomorphic to a subgroup of the Hawaiian earring group under specific conditions.
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.
Characterizes group connections on group bundles.
problem Understanding connections on group bundles.
method Characterizes connections as affine spaces and uses the Ambrose-Singer theorem.
result Group connections form an affine space over cocycles.
The paper defines Dirac structures on connection spaces and their properties.
problem Defining Dirac structures on spaces of connections.
method Twisted Dirac structures on spaces of irreducible connections over manifolds, described by the Cartan 3-form.
result Spaces of flat connections are endowed with Dirac structures, and their properties are discussed.
Defines semi-symmetric metric connection on super warped products.
problem Computing curvature and Ricci tensors on super warped products.
method Introduced semi-symmetric metric connection and conditions for Einstein spaces.
result Conditions for super warped product spaces to be Einstein with semi-symmetric metric connection.
The paper classifies invariant connections and Einstein structures on isotropy irreducible spaces.
problem Classifying invariant connections and Einstein structures on isotropy irreducible spaces.
method Systematic study and classification of invariant affine or metric connections on naturally reductive spaces.
result Classification of invariant metric connections with skew-torsion and abla-Einstein structures. Study classifies Einstein-Yang-Mills spaces in 4D symmetric spaces.
problem Classifying Lorentzian symmetric spaces with Einstein-Yang-Mills properties.
method Classification based on invariant metric connections and diagonal metrics.
result Four-dimensional symmetric spaces with nontrivial isotropy groups are classified.
The kth finite subset space of a topological space X is the space exp_k X of non-empty subsets of X of size at most k, topologised as a quotient of X^k. Using results from our earlier paper (math.GT/0210315) on the finite subset spaces of connected graphs we show that the kth finite subset space of a connected cell com…
The paper studies four Hermitian structures on twistor spaces.
problem Understanding the geometry of Hermitian surfaces via twistor spaces.
method Defining four almost Hermitian structures on twistor spaces using canonical connections.
result Relations between twistor space geometry and Hermitian surface geometry.
The paper studies special warped products with a specific connection on super Riemannian manifolds.
problem Investigating curvature and Ricci tensors on super warped product spaces with a semi-symmetric non-metric connection.
method Defined a semi-symmetric non-metric connection, computed curvature and Ricci tensors, and introduced and analyzed two types of super warped product spaces.
result Conditions for two super warped product spaces with a semi-symmetric non-metric connection to be Einstein spaces are provided.
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.
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.
Affine connections become simple spaces locally.
problem Understanding affine connections and their properties.
method Combinatorial and geometric argument.
result Affine connections are locally affine spaces.
We consider local invariants of general connections (with torsion). The group of origin-preserving diffeomorphisms acts on a space of jets of general connections. Dimensions of moduli spaces of generic connections are calculated. Poincaré series of the geometric structure of connection is constructed, and shown to be a…
The action of origin-preserving diffeomorphisms on a space of jets of symmetric connections is considered. Dimensions of moduli spaces of generic connections are calculated. Poincaré series of the geometric structure of symmetric connection is constructed, and shown to be a rational function.
The paper characterizes surfaces in 4D space forms with flat normal connection.
problem Characterizing surfaces in 4D space forms with specific geometric properties.
method Analyzing linearly dependent conditions and using properties of sectional curvature.
result Characterizations of space-like and time-like surfaces with flat normal connection.
Study on submanifolds in specific geometric spaces.
problem Analysis of submanifolds in generalized Sasakian-space-forms.
method Examines almost semi-invariant submanifolds with various connections.
result Provides new results on submanifolds with respect to different connections.
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.
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.
The paper shows a 3D shape can't fit into certain compact spaces.
problem Whether a specific 3D shape can fit into compact spaces.
method Using techniques from Sternfeld's thesis, the paper proves the impossibility.
result The contractible open 3-manifold cannot embed in compact, locally connected and locally 1-connected metric spaces.
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. The paper constructs a Hitchin connection for a broad class of Kähler structures.
problem Developing a Hitchin connection for a large class of Kähler structures.
method Generalizing previous work, the paper constructs a Hitchin connection on the space of compatible complex structures on arbitrary symplectic manifolds.
result The constructed Hitchin connection is a partial connection on the tangent space of compatible complex structures, defined in a neighborhood of natural families of complex structures.
Study classifies moduli spaces of spin connections on 3D homogeneous spaces.
problem Classifying moduli spaces of spin connections on 3D homogeneous spaces.
method Analysis of the topology of moduli spaces, focusing on finite-dimensional topological manifolds with trivial homotopy groups.
result Moduli spaces are finite-dimensional topological manifolds with trivial homotopy groups, essential for consistent cosmological models.
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…
Research shows RCD* spaces are semi-locally simply connected.
problem Understanding the topological properties of RCD* spaces.
method Proving semi-locally simply connected property for any point and radius.
result RCD* spaces are semi-locally simply connected.
Reinterprets Schrödinger equation using Cartan connection for geometric investigation.
problem Investigating the geometry of the space for Schrödinger equation solutions.
method Constructs Cartan connection from scaling Lie-Bäcklund group on jet space.
result Demonstrates a new geometric approach to Schrödinger equation.
In this paper, we study CAT(0) spaces with non-locally connected boundary. We give some condition of a CAT(0) space whose boundary is not locally connected.
Instantons on ALF spaces constructed from bow data.
problem Constructing instantons on Asymptotically Locally Flat spaces.
method Using bow data to represent ALF spaces and their moduli spaces, constructing anti-self-dual connections.
result Anti-self-dual connections on ALF spaces are instantons with finite action.
The paper investigates compatible linear connections on Randers spaces and finds a unique extremal connection.
problem Investigating compatible linear connections on Randers spaces.
method Transformed compatibility equations by taking torsion components as variables and determined when these equations have solutions.
result Characterized Randers spaces as non-Riemannian generalized Berwald spaces with a positive constant norm of perturbing term.
Genus 3 Heegaard groups of lens space connected sums are finitely generated.
problem Understanding the structure of mapping class groups of genus 3 Heegaard splittings.
method Proved finitely generated property through connected reducing sphere complexes.
result Mapping class groups are finitely generated and complexes are connected.
For a system of second order differential equations we determine a nonlinear connection that is compatible with a given generalized Lagrange metric. Using this nonlinear connection, we can find the whole family of metric nonlinear connections that can be associated with a system of SODE and a generalized Lagrange struc…
In this paper, we describe the space of adapted connections on a metric contact manifold through the space of their torsion tensors. The torsion tensor is an element of the space of TM-valued two-forms, which splits into various subspaces. We study the parts of the torsion tensor according to this splitting to complete…
We extend the result in J. Reine Angew. Math. 664, 29-53, to the non-compact case. Namely, we prove that the canonical connection on a simply connected and irreducible naturally reductive space is unique, provided the space is not a sphere, a compact Lie group with a bi-invariant metric or its symmetric dual. In partic…
SU(2) flat connections link to 3D geometry with cosmological constant.
problem Mapping flat connections on Riemann surfaces to 3D twisted geometry.
method Relating flat connection quantities to geometrical quantities in discrete 3D space.
result Moduli space of SU(2) flat connections generalizes phase space of twisted geometry.
Maximal isometries define canonical connections in sub-Riemannian spaces.
problem Characterizing sub-Riemannian spaces with maximal isometry groups.
method Study of sub-Riemannian spaces with maximal isometry groups and canonical connections.
result Canonical connections generalize Levi-Civita connections and have more invariants.
Odd m-fold connected sums of complex projective spaces admit almost complex structures.
problem Existence of almost complex structures on connected sums of complex projective spaces.
method Analyzing the m-fold connected sum m#CP2n for m odd or even. result Only odd m-fold connected sums of complex projective spaces admit almost complex structures.
Quantum connections replace metrics with operator inner products.
problem Quantifying geometric properties in quantum systems.
method Defining quantum connections and duals using operator fields and inner products.
result Holonomy and dual connections are equivalent in quantum geometry.
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.
Study biharmonic hypersurfaces in Sasakian space form using Tanaka-Webster connection.
problem Exploring biharmonic hypersurfaces in Sasakian space form.
method Using Tanaka-Webster connection to study biharmonic hypersurfaces.
result Developed new insights into biharmonic hypersurfaces in Sasakian space form.
Characterizes homogeneous spaces with geometric structures using connections.
problem Characterizing homogeneous spaces with various geometric structures.
method Using connections to characterize reductive homogeneous spaces.
result Generalizes Ambrose-Singer theorem to non-Riemannian geometries.
Study invariant connections on multivariate Gaussian distributions.
problem Understanding statistical connections on multivariate Gaussian distributions.
method Investigate invariant connections on N0n with the Fisher metric. result Explicitly determined invariant connections and their moduli spaces.
Knots formed from torus knots are not concordant to L-space knots.
problem Understanding concordance in knots formed from connected sums of torus knots.
method Analyzing properties of knots formed from connected sums of torus knots.
result Knots formed from torus knots are not concordant to L-space knots.
We introduce a symplectic structure on the space of connections in a G-principal bundle over a four-manifold and the Hamiltonian action on it of the group of gauge transformations which are trivial on the boundary. The symplectic reduction becomes the moduli space of flat connections over the manifold. On the moduli sp…
Study special affine connections on symmetric spaces and their products.
problem Characterize special affine connections on symmetric spaces.
method Analyze canonical affine connections, introduce special products, and study holonomy Lie algebras.
result Established a correspondence between special affine connections and special products on the Lie algebra.
Classifies 3D spaces using specific invariants.
problem Classifying 3D simply connected Mori fibre spaces.
method Uses finitely many numerical invariants.
result Finds finitely many invariants to classify diffeomorphism types.