Canonical gravity can be formulated by means of a densitized dreibein together with an SU(2) connection. These so-called Ashtekar variables are the fundamental quantities, loop quantum gravity is resting on. In this paper we review these variables from the perspective of fibre bundles. This is straightforward for the d…
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48 results for “Ashtekar connection”
It was pointed out that the space of hermitian triples is an anology of the hermitian connection space. Generalizing the Ashtekar - Isham procedure one can quantize the space of hermitian triples as well as the original one. Here we add an example how this similarity can be exploited in a quantum theory of riemannian g…
This work interprets supergravity as a super Cartan geometry linking it to Yang-Mills theory.
problem Geometric interpretation of supergravity and its relation to Yang-Mills theory.
method Using enriched categories and super Cartan geometries, the authors link supergravity to Yang-Mills theory.
result Non-extended D=4 supergravity naturally arises in this framework.
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 investigate the differential calculus defined by Ashtekar and Lewandowski on projective limits of manifolds by means of cylindrical smooth functions and compare it with the C^infty calculus proposed by Froehlicher and Kriegl in more general context. For products of connected manifolds, a Boman theorem is proved, sho…
In the present paper a global conformal invariant of a closed initial data set is constructed. A spacelike hypersurface in a Lorentzian spacetime naturally inherits from the spacetime metric a differentiation , the so-called real Sen connection, which turns out to be determined completely by the ini…
We solve Einstein vacuum equations in a spacetime region up to the "center" of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smoo…
We prove two theorems, announced in hep-th/0108170, for static spacetimes that solve Einstein's equation with negative cosmological constant. The first is a general structure theorem for spacetimes obeying a certain convexity condition near infinity, analogous to the structure theorems of Cheeger and Gromoll for manifo…
New spaces at infinity identified for Minkowski spacetime.
problem Characterizing asymptotic infinities of Minkowski spacetime.
method Embedding and describing homogeneous spaces of the Poincaré group.
result Determined new structures on asymptotic infinities.
A new connection in Finsler geometry unifies various types of connections.
problem Introducing a unified connection in Finsler geometry.
method Using the pullback formalism, a new linear connection is introduced and investigated.
result The existence and uniqueness of the new connection are proved intrinsically.
A -metric manifold has an almost complex or almost product structure and a compatible metric . We show that there exists a canonical involution in the set of connections on such a manifold, which allows to define a projection over the set of connections adapted to . This projection sends the Le…
We compute all the simply connected homogeneous and infinitesimally homogeneous surfaces admitting one or more invariant affine connections. We find exactly six non equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces ad…
Canonical connections and algebraic Ricci solitons of three-dimensional Lorentzian Lie groupsmath.DG
The paper classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying algebraic Ricci solitons on three-dimensional Lorentzian Lie groups.
method Computed canonical and Kobayashi-Nomizu connections and their curvatures; defined algebraic Ricci solitons.
result Classified algebraic Ricci solitons on specific Lorentzian Lie groups.
New normalization condition for sub-Riemannian connections.
problem Normalizing connections on sub-Riemannian manifolds.
method Formulated in terms of Cartan connections, depends on curvature's first degree of homogeneity.
result A compatible partial affine connection can be uniquely extended to a full affine connection and a grading of the tangent bundle.
Odd connections on supermanifolds are defined and their properties studied.
problem Defining and understanding odd quasi-connections on supermanifolds.
method Examined odd quasi-connections, defined torsion and curvature, and identified special classes.
result Odd connections on supermanifolds are shown to have torsion and curvature tensors.
The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.
problem Classifying Lorentzian Lie groups based on specific tensor properties.
method Classification of three-dimensional Lorentzian Lie groups based on Ricci tensors and quasi-statistical structures associated with different affine connections.
result The paper classifies three-dimensional Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures associated with Bott, canonical, and Kobayashi-Nomizu connections.
The paper proves monotonicity formulas for minimal connections and their applications.
problem Understanding critical points of volume functionals in Riemannian geometry.
method Developed monotonicity formulas for minimal connections under specific conditions.
result Established vanishing theorems for minimal connections on Euclidean spaces and dDT connections on G2-manifolds.
The paper solves the Integration Problem for principal connections.
problem Describing discrete connections associated with a principal connection.
method Using the Lie or derivative functor to induce connections on the principal bundle.
result For flat principal connections, the Integration Problem has a unique solution among flat discrete connections.
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 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.
In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connect…
Develops torsion dual connections for statistical manifolds.
problem Defining statistical manifolds using dual connections.
method Introduces torsion dual connections and proves their properties.
result Curvature tensor of torsion dual connections has specific divergence.
Extends connections on Lie groupoids, proving completeness conditions.
problem Existence and completeness of multiplicative connections on Lie groupoid fibrations.
method Introduces and investigates multiplicative Ehresmann connections on Lie groupoid fibrations.
result Conditions for completeness of multiplicative connections on Lie groupoid fibrations.
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.
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.
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.
Sprays on Frechet manifolds connect connections and tangent structures.
problem Characterizing linear symmetric connections on Frechet manifolds.
method Constructing connection maps and linear symmetric connections on tangent and second-order tangent bundles using sprays.
result A bijective correspondence exists between linear symmetric connections on tangent bundles and sprays.
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.
Defines a new natural connection on Riemannian Π-manifolds.
problem Characterizing natural connections on Riemannian Π-manifolds.
method Introducing and analyzing the first natural connection with torsion.
result Relations between the first natural connection and Levi-Civita connection are established.
In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.
This paper explores how local behavior of meromorphic connections on the projective line determines the global connection.
problem Determining the global meromorphic connection based on specified local behavior at singular points.
method Expository discussion of various problems related to meromorphic connections with specified local behavior, including Deligne-Simpson and rigidity problems.
result The existence and nonemptiness of moduli spaces of meromorphic connections with specified local behavior.
Paper extends Simons theorem to -Yang-Mills connections for instability.
problem Tackles instability of -Yang-Mills connections.
method Extends Simons theorem to -Yang-Mills connections using Kobayashi-Ohnita-Takeuchi's method.
result Derives a sufficient condition for instability of non-flat -Yang-Mills connections.
Recently the present authors introduced a general class of Finsler connections which leads to a smart representation of connection theory in Finsler geometry and yields to a classification of Finsler connections into the three classes. Here the properties of one of these classes namely the Berwald-type connections whic…
The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine connections with slowly decaying positive Ricci curvature.
Flat Yang-Mills connections on pinched manifolds.
problem Stability of Yang-Mills connections on compact manifolds.
method Pinching conditions and weak stability criteria.
result No non-flat weakly stable Yang-Mills connections on δ(n)-pinched compact simply-connected Riemannian manifolds.
The study classifies holomorphic projective connections on complex threefolds.
problem Characterizing holomorphic projective connections on complex threefolds.
method Analyzing properties of holomorphic projective connections on complex projective threefolds.
result Holomorphic projective connections on complex threefolds are either flat or translation invariant on abelian threefolds.
In the present work, we introduce a linear connection (preserving the almost product structure and the Riemannian metric) on Riemannian almost product manifolds. This connection, called P-connection, is an analogue of the first canonical connection of Lichnerowicz in the Hermitian geometry and the B-connection in the g…
We assume a vector bundle with a general linear connection and a classical linear connection $\Lam$ on . We prove that all classical linear connections on the total space naturally given by $(\Lam, K)$ form a 15-parameter family. Further we prove that all connections on naturally given by…
Classifies Ricci solitons on specific Lorentzian Lie groups.
problem Classifying Ricci solitons on three-dimensional Lorentzian Lie groups.
method Examined canonical and perturbed canonical connections, as well as Kobayashi-Nomizu connections and perturbed Kobayashi-Nomizu connections.
result Classified affine Ricci solitons on three-dimensional Lorentzian Lie groups with product structure.
Paper develops a unified framework for Lie algebroid connections on various bundles.
problem Unified framework for Lie algebroid connections on vector and principal bundles.
method Generalized Atiyah algebroid structure and its short exact sequence.
result Explicit constructions of Atiyah classes for Lie algebroid connections.
Characterizes connections on multivariate normal distributions.
problem Characterizing connections on statistical manifold of multivariate normal distributions.
method Analyzes statistical manifold of multivariate normal distributions.
result The Amari-Chentsov connection is characterized by conjugate symmetry.
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.
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 -metric manifolds. We characterize when such a connection is a…
Holonomy of Weyl connections in Lorentzian space classified.
problem Classifying holonomy algebras of Weyl connections in Lorentzian signature.
method Classification through construction of examples of Weyl connections with all possible holonomy algebras.
result Examples of Weyl connections with all possible holonomy algebras constructed.
We define what we call morphisms of Cartan connections. We generalize the main theorems on Cartan connections to theorems on morphisms. Many of the known constructions involving Cartan connections turn out to be examples of morphisms. We prove some basic results concerning completeness of Cartan connections. We provide…
Universal connection constructed using diffeology theory.
problem Natural connection on bundles of paths on manifolds.
method Diffeological construction of Singer's universal connection.
result Functorial equivalence between holonomy categories and diffeological bundle-connection pairs.
Curvature without connectionmath.DG
New theory connects geometry without relying on connections.
problem Developing geometric structures without connections.
method Proposed alternative approach to Lie groups and geometric structures.
result Geometric structures can be developed independently of connections.
The present paper deals with some results of submanifolds of generalized Sasakian-space-forms in \cite{ALEGRE3} with respect to semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection and Tanaka-webster connection.