Einstein's Connection Research Articles

Automorphism group of principal bundles, Levi reduction and invariant connections

Let M be a compact connected complex manifold and G a connected reductive complex affine algebraic group. Let $$E_G$$ be a holomorphic principal G–bundle over M and $$T\, \subset \, G$$ a torus containing the connected component of the center of G. Let N (respectively, C) be the normalizer (respectively, centralizer) of T in G, and let W be the Weyl group N / C for T. We prove that there is a natural bijective correspondence between the following two: The composition of maps $$E'_C\, {\mathop {\longrightarrow }\limits ^{\psi }}\, E_W \, {\mathop {\longrightarrow }\limits ^{\phi }}\, M$$ defines a principal N–bundle on M. This principal N–bundle $$E_N$$ is a reduction in structure group of $$E_G$$ to N. Given a complex connection $$\nabla $$ on $$E_G$$ , we give a necessary and sufficient condition for $$\nabla $$ to be induced by a connection on $$E_N$$ . This criterion relates Hermitian–Einstein connections on $$E_G$$ and $$E'_C$$ in a very precise manner.

The study on the Einstein's connection in $5$-dimensional $ES$-manifold for the second class

The manifold $ {}^*{g} - ESX_n $ is a generalized $ n $-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $ {}^*{g}^{ \lambda \nu } $ through the $ ES $-connection which is both Einstein and semi-symmetric. The purpose of the present paper is to derive a new set of powerful recurrence relations and to prove a necessary and sufficient condition for a unique Einstein's connection to exist in $5$-dimensional ${}^*{g}-ESX_5$ and to display a surveyable tnesorial representation of $5$-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations in the second class.

Einstein's connection in $5$-dimensional $ES$-manifold

The manifold $ {}^*{g} - ESX_n $ is a generalized $ n $-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $ {}^*{g}^{ \lambda \nu } $ through the $ ES $-connection which is both Einstein and semi-symmetric. The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in $5$-dimensional ${}^*{g}-ESX_5$ and to display a surveyable tnesorial representation of $5$-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations in the first class.

Generalized Lagrangian mean curvature flows: The cotangent bundle case

Abstract In [18], we defined a generalized mean curvature vector field on any almost Lagrangian submanifold with respect to a torsion connection on an almost Kähler manifold. The short time existence of the corresponding parabolic flow was established. In addition, it was shown that the flow preserves the Lagrangian condition as long as the connection satisfies an Einstein condition. In this article, we show that the canonical connection on the cotangent bundle of any Riemannian manifold is an Einstein connection (in fact, Ricci flat). The generalized mean curvature vector on any Lagrangian submanifold is related to the Lagrangian angle defined by the phase of a parallel ( n , 0 ) {(n,0)} -form, just like the Calabi–Yau case. We also show that the corresponding Lagrangian mean curvature flow in cotangent bundles preserves the exactness and the zero Maslov class conditions. At the end, we prove a long time existence and convergence result to demonstrate the stability of the zero section of the cotangent bundle of spheres.

EINSTEIN'S CONNECTION IN 3-DIMENSIONAL ES-MANIFOLD

The manifold $ {}^*{g} - ESX_n $ is a generalized $ n $-dimensional Riemannian manifold on which the differential geometric structure is imposed by the unified field tensor $ {}^*{g}^{ \lambda \nu } $ through the $ ES $-connection which is both Einstein and semi-symmetric. The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in $3$-dimensional ${}^*{g}-ESX_3$ and to display a surveyable tnesorial representation of $3$-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations in the first class.

THE YANG–MILLS FUNCTIONAL AND BOGOMOLOV INEQUALITY FOR ARBITRARY PRINCIPAL BUNDLES OVER KÄHLER MANIFOLDS

We study the Yang–Mills functional for principal fiber bundles with structure group a compact Lie group K over a Kähler manifold. In particular, we analyze the absolute minimizers for this functional and prove that they are exactly the Einstein K-connections. By means of the structure of the Yang–Mills functional at an absolute minimum, we prove that the characteristic classes of a principal K-bundle which admits an Einstein connection satisfy two inequalities. One of them is a generalization of the Bogomolov inequality whereas the other is an inequality related to the center of the structure group. Therefore, this way we offer a new and natural proof of the Bogomolov inequality that helps understanding its origin. Finally, in view of the Hitchin–Kobayashi correspondence we prove that every (poly-)stable principal Kℂ-bundle has to satisfy this generalized Bogomolov type inequality.

Approximate Hermitian–Einstein connections on principal bundles over a compact Riemann surface

Let \(X\) be a compact connected Riemann surface and \(G\) a connected reductive complex affine algebraic group. Given a holomorphic principal \(G\)-bundle \(E_G\) over \(X\), we construct a \(C^\infty \) Hermitian structure on \(E_G\) together with a \(1\)-parameter family of \(C^\infty \) automorphisms \(\{F_t\}_{t\in \mathbb R }\) of the principal \(G\)-bundle \(E_G\) with the following property: Let \(\nabla ^t\) be the connection on \(E_G\) corresponding to the Hermitian structure and the new holomorphic structure on \(E_G\) constructed using \(F_t\) from the original holomorphic structure. As \(t\rightarrow -\infty \), the connection \(\nabla ^t\) converges in \(C^\infty \) Frechet topology to the connection on \(E_G\) given by the Hermitian–Einstein connection on the polystable principal bundle associated to \(E_G\). In particular, as \(t\rightarrow -\infty \), the curvature of \(\nabla ^t\) converges in \(C^\infty \) Frechet topology to the curvature of the connection on \(E_G\) given by the Hermitian–Einstein connection on the polystable principal bundle associated to \(E_G\). The family \(\{F_t\}_{t\in \mathbb R }\) is constructed by generalizing the method of [6]. Given a holomorphic vector bundle \(E\) on \(X\), in [6] a \(1\)-parameter family of \(C^\infty \) automorphisms of \(E\) is constructed such that as \(t\rightarrow -\infty \), the curvature converges, in \(C^0\) topology, to the curvature of the Hermitian–Einstein connection of the associated graded bundle.

HERMITIAN–EINSTEIN CONNECTIONS ON PRINCIPAL BUNDLES OVER FLAT AFFINE MANIFOLDS

Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle EG over M admits a Hermitian–Einstein structure if and only if EG is polystable. A polystable flat principal G-bundle over M admits a unique Hermitian–Einstein connection. We also prove the existence and uniqueness of a Harder–Narasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler.

Hermitian–Einstein connections on polystable orthogonal and symplectic parabolic Higgs bundles

Let X be a smooth complex projective curve and S⊂X a finite subset. We show that an orthogonal or symplectic parabolic Higgs bundle on X with parabolic structure over S admits a Hermitian–Einstein connection if and only if it is polystable.

Determinant line bundle on moduli space of parabolic bundles

In Biswas and Raghavendra (Proc Indian Acad Sci (Math Sci) 103:41–71, 1993; Asian J Math 2:303–324, 1998), a parabolic determinant line bundle on a moduli space of stable parabolic bundles was constructed, along with a Hermitian structure on it. The construction of the Hermitian structure was indirect: The parabolic determinant line bundle was identified with the pullback of the determinant line bundle on a moduli space of usual vector bundles over a covering curve. The Hermitian structure on the parabolic determinant bundle was taken to be the pullback of the Quillen metric on the determinant line bundle on the moduli space of usual vector bundles. Here a direct construction of the Hermitian structure is given. For that we need to establish a version of the correspondence between the stable parabolic bundles and the Hermitian–Einstein connections in the context of conical metrics. Also, a recently obtained parabolic analog of Faltings’ criterion of semistability plays a crucial role.

Einstein connections and involutions via parabolic geometries

This paper demonstrates a way of finding Einstein connections (affine connections whose Ricci tensor is non-degenerate and covariantly constant) via parabolic geometry. Extending the results in the projective and conformal cases, it demonstrates that the existence of a preserved involution σ on an associated bundle A leads–under mild conditions–to the construction of a specific Einstein connection, among the Weyl connections of the parabolic geometry. The conditions necessary for the existence of such ‘Einstein involutions’ are then presented, corresponding to a small family of holonomy reductions for the Cartan connection defining the parabolic geometry. This is illustrated with a few examples.

On the eigenvalues of the twisted Dirac operator

Given a compact Riemannian spin manifold whose untwisted Dirac operator has trivial kernel, we find a family of connections ∇At for t∊[0,1] on a trivial vector bundle of rank no larger than dim M+1, such that the first eigenvalue of the twisted Dirac operator DAt is nonzero for t≠1 and vanishes for t=1. However, if one restricts the class of twisting connections considered, then nonzero lower bounds do exist. We illustrate this fact by establishing a nonzero lower bound for the Dirac operator twisted by Hermitian–Einstein connections over Riemann surfaces.

On the spectrum of the twisted Dolbeault Laplacian over Kähler manifolds

We use Dirac operator techniques to a establish sharp lower bound for the first eigenvalue of the Dolbeault Laplacian twisted by Hermitian–Einstein connections on vector bundles of negative degree over compact Kähler manifolds.

EIGHT-DIMENSIONAL EINSTEIN'S CONNECTION FOR THE FIRST CLASS II. THE EINSTEIN'S CONNECTION IN 8-g-UFT

Lower dimensional cases of Einstein's connection were already investigated by many authors for n = 2, 3, 4, 5, 6, 7. In the following series of two papers, we present a surveyable tensorial representation of 8-dimensional Einstein's connection in terms of the unified field tensor: I. The recurrence relations in 8-g-UFT II. The Einstein's connection in 8-g-UFT In our previous paper [1], we investigated some algebraic structure in Einstein's 8-dimensional unified field theory (i.e., 8-g-UFT), with emphasis on the derivation of the recurrence relations of the third kind which hold in 8-g-UFT. This paper is a direct continuation of [1]. The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein's connection to exist in 8-g-UFT and to display a surveyable tensorial representation of 8-dimensional Einstein's connection in terms of the unified field tensor, employing the powerful recurrence relations of the third kind obtained in the first paper [1]. All considerations in this paper are restricted to the second class only of the generalized 8-dimensional Riemannian manifold , since the case of the first class are done in [2], [3] and the case of the third class, the simplest case, was already studied by many authors.

Kähler structure on moduli spaces of principal bundles

Let M be a moduli space of stable principal G-bundles over a compact Kähler manifold ( X , ω X ) , where G is a reductive linear algebraic group defined over C . Using the existence and uniqueness of a Hermite–Einstein connection on any stable G-bundle P over X, we have a Hermitian form on the harmonic representatives of H 1 ( X , ad ( P ) ) , where ad ( P ) is the adjoint vector bundle. Using this Hermitian form a Hermitian structure on M is constructed; we call this the Petersson–Weil form. The Petersson–Weil form is a Kähler form, a fact which is a consequence of a fiber integral formula that we prove here. The curvature of the Petersson–Weil Kähler form is computed. Some further properties of this Kähler form are investigated.

Field equations onSE(k)-manifoldX n

An Einstein connection which is both a special connection and a (k)-connection is called anSE(k)-connection. And a generalized even-dimensional Riemannian manifoldXn with the so-called “SE(k)-condition” defined by theSE(k)-connection is called theSE(k)-manifold. We obtain the necessary and sufficient condition that there is a uniqueSE(k)-connection inXn. Next, using these results, we define theSE(k)-manifold and study the properties of the curvature tensors and the field equations in theSE(k)-manifoldXn.

Three- and five-dimensional considerations of the geometry of Einstein's* g-unified field theory

We study the algebraic and differential geometric structures of three- and five-dimensional* g-unified field theory, with emphasis on the five-dimensional* g-unified field theory, in which we derive a new set of powerful recurrence relations which hold in a five-dimensional generalized Riemannian manifoldX 5 , prove a necessary and sufficient condition for the existence and uniqueness of the solution of the system of the Einstein equations in the first two classes, and find a precise tensorial representation of the Einstein connection Γ λ in terms of*gνν .

Some recurrence relations and Einstein's connection in 2-dimensional unified field theory

Some recurrence relations and Einstein's connection in 2-dimensional unified field theory

Paths of spinning particles in general relativity as geodesics of an Einstein connection

This paper deals with the Papapetrou-Pirani equations of motion for a spinning test particle in general relativity. The motion of the center of mass can be represented by the geodesic equation of an affine connection that is the sum of the Christoffel connection and a tensor that depends on the Riemann-Christoffel curvature tensor, the mass of the particle, its 4-velocity, and its spin tensor. The connection is not unique, and here it is chosen to satisfy one of the basic geometrical principles of Einstein's unified field theory: The symmetric part of the fundamental tensor of the geometry is specified to be the metric tensor of general relativity. The special case of conformally flat space-times is discussed.

Electromagnetism and Geodesics of an Einstein Connection

Electromagnetism and Geodesics of an Einstein Connection