Invariant Connections Research Articles

Parabolic symmetric spaces

We study here systems of symmetries on $|1|$--graded parabolic geometries. We are interested in smooth systems of symmetries and we discuss non--flat homogeneous $|1|$--graded geometries. We show the existence of an invariant admissible affine connection under quite weak condition on the system.

Invariant Yang–Mills connections over non-reductive pseudo-Riemannian homogeneous spaces

We study invariant gauge fields over the 4-dimensional non-reductive pseudo-Riemannian homogeneous spaces G/K recently classified by Fels & Renner (2006). Given H compact semi-simple, classification results are obtained for principal H-bundles over G/K admitting: (1) a G-action (by bundle automorphisms) projecting to left multiplication on the base, and (2) at least one G-invariant connection. There are two cases which admit nontrivial examples of such bundles and all G-invariant connections on these bundles are Yang-Mills. The validity of the principle of symmetric criticality (PSC) is investigated in the context of the bundle of connections and is shown to fail for all but one of the Fels-Renner cases. This failure arises from degeneracy of the scalar product on pseudo-tensorial forms restricted to the space of symmetric variations of an invariant connection. In the exceptional case where PSC is valid, there is a unique G-invariant connection which is moreover universal, i.e. it is the solution of the Euler-Lagrange equations associated to any G-invariant Lagrangian on the bundle of connections. This solution is a canonical connection associated with a weaker notion of reductivity which we introduce.

Invariant connections on Euclidean space

We recall and solve the equivalence problem for a flat C^1 connection ∇ in Euclidean space, with methods from the theory of differential equations. The problem consists in finding an affine transformation of R^n taking ∇ to the so called trivial connection. Generalized solutions are found in dimension 1 and some example problems are solved in dimension 2, mainly dealing with flat connections. A description of invariant connections in the plane is attempted, in view the study of real orbifolds. Complex meromorphic connections are shown in the cone cL(p, q) of a lens-space.

The conformal Killing equation on forms—prolongations and applications

We construct a conformally invariant vector bundle connection such that its equation of parallel transport is a first order system that gives a prolongation of the conformal Killing equation on differential forms. Parallel sections of this connection are related bijectively to solutions of the conformal Killing equation. We construct other conformally invariant connections, also giving prolongations of the conformal Killing equation, that bijectively relate solutions of the conformal Killing equation on k-forms to a twisting of the conformal Killing equation on ( k − ℓ ) -forms for various integers ℓ. These tools are used to develop a helicity raising and lowering construction in the general setting and on conformally Einstein manifolds.

PONTRJAGIN FORMS AND INVARIANT OBJECTS RELATED TO THE Q-CURVATURE

It was shown by Chern and Simons that the Pontrjagin forms are conformally invariant. We show them to be the Pontrjagin forms of the conformally invariant tractor connection. The Q-curvature is intimately related to the Pfaffian. Working on even-dimensional manifolds, we show how the k-form operators Qkof [12], which generalize the Q-curvature, retain a key aspect of the Q-curvature's relation to the Pfaffian, by obstructing certain representations of natural operators on closed forms. In a closely related direction, we show that the Qkgive rise to conformally invariant quadratic forms Θkon cohomology that interpolate, in a suitable sense, between the integrated metric pairing (at k = n/2) and the Pfaffian (at k = 0). Using a different construction, we show that the Qkoperators yield a map from conformal structures to Lagrangian subspaces of the direct sum Hk⊕ Hk(where Hkis the dual of the de Rham cohomology space Hk); in an appropriate sense this generalizes the period map. We couple the Qkoperators with the Pontrjagin forms to construct new natural densities that have many properties in common with the original Q-curvature; in particular these integrate to global conformal invariants. We also work out a relevant example, and show that the proof of the invariance of the (nonlinear) action functional whose critical metrics have constant Q-curvature extends to the action functionals for these new Q-like objects. Finally we set up eigenvalue problems that generalize to Qk-operators the Q-curvature prescription problem.

Nonholonomic connections with Galilean groups of transformations

We construct second-order nonholonomic smooth invariant connections in the case of Galilean G(1,n) group representation in the canonical fiber bundle π: R n +1 → R n .

ON A FINSLER SPACE WITH (α, β)-METRIC AND CERTAIN METRICAL NON-LINEAR CONNECTION

The purpose of this paper is to introduce an L-metrical non-linear connection <TEX>$N_j^{*i}$</TEX> and investigate a conformal change in the Finsler space with <TEX>$({\alpha},\;{\beta})-metric$</TEX>. The (v)h-torsion and (v)hvtorsion in the Finsler space with L-metrical connection <TEX>$F{\Gamma}^*$</TEX> are obtained. The conformal invariant connection and conformal invariant curvature are found in the above Finsler space.

Invariant noncommutative connections

In this paper we classify invariant noncommutative connections in the framework of the algebra of endomorphisms of a complex vector bundle. It has been proven previously that this noncommutative algebra generalizes in a natural way the ordinary geometry of connections. We use explicitly some geometric constructions usually introduced to classify ordinary invariant connections, and we expand them using algebraic objects coming from the noncommutative setting. The main result is that the classification can be performed using a “reduced” algebra, an associated differential calculus and a module over this algebra.

A geometric approach to scalar field theories on the supersphere

Following a strictly geometric approach we construct globally supersymmetric scalar field theories on the supersphere, defined as the quotient space S2∣2=UOSp(1∣2)∕U(1). We analyze the superspace geometry of the supersphere, in particular deriving the invariant vielbein and spin connection from a generalization of the left-invariant Maurer–Cartan form for Lie groups. Using this information we proceed to construct a superscalar field action on S2∣2, which can be decomposed in terms of the component fields, yielding a supersymmetric action on the ordinary two-sphere. We are able to derive Lagrange equations and Noether’s theorem for the superscalar field itself.

Covariant and equivariant formality theorems

We give a proof of Kontsevich's formality theorem for a general manifold using Fedosov resolutions of algebras of polydifferential operators and polyvector fields. The main advantage of our construction of the formality quasi-isomorphism is that it is based on the use of covariant tensors unlike Kontsevich's original proof, which is based on ∞-jets of polydifferential operators and polyvector fields. Using our construction we prove that if a group G acts smoothly on a manifold M and M admits a G -invariant affine connection then there exists a G -equivariant quasi-isomorphism of formality. This result implies that if a manifold M is equipped with a smooth action of a finite or compact group G or equipped with a free action of a Lie group G then M admits a G -equivariant formality quasi-isomorphism. In particular, this gives a solution of the deformation quantization problem for an arbitrary Poisson orbifold.

The First-Order Nonholonomic Connections with the Galilean Groups of Local Transformations. III

In the canonical smooth fiber bundles $$\prod :\mathbb{R}^{n + 1} \to \mathbb{R}^n $$ endowed with the metric tensor fields of relevant structure, we consider natural representations of the Galilean groups $$\mathbb{G}$$ (1, n) and construct $$\mathbb{G}$$ (1, n)-invariant generalized differential-geometric connections. In both regular and special cases of the representations of the considered groups $$\mathbb{G}$$ (1, n), we find all affine nonholonomic $$\Gamma _{1^ - } ,\Gamma _{2^ - } $$ , and Γ1,2-connections of the first order (see [1]–[3]) possessing the local Lie groups of transformations $$\mathbb{G}$$ (1, n) and also describe the corresponding $$\mathbb{G}$$ (1, n)invariant planar connections.

Tau-functions on Hurwitz Spaces

We construct a flat holomorphic line bundle over a connected component of the Hurwitz space of branched coverings of the Riemann sphere P1. A flat holomorphic connection defining the bundle is described in terms of the invariant Wirtinger projective connection on the branched covering corresponding to a given meromorphic function on a Riemann surface of genus g. In genera 0 and 1 we construct a nowhere vanishing holomorphic horizontal section of this bundle (the ‘Wirtinger tau-function’). In higher genus we compute the modulus square of the Wirtinger tau-function. In particular one gets formulas for the isomonodromic tau-functions of semisimple Frobenius manifolds connected with the Hurwitz spaces Hg,N(1,⋯,1).

CONFORMAL CHANGES OF A RIZZA MANIFOLD WITH A GENERALIZED FINSLER STRUCTURE

We are devoted to dealing with the conformal the- ory of a Rizza manifold with a generalized Finsler metric Gij(x;y) and a new conformal invariant non-linear connection M i j(x;y) con- structed from the generalized Chern's non-linear connection N i j(x; y) and almost complex structure f i j(x). First, we find a confor- mal invariant connection (Mj i k;M i j;Cj i k) and conformal invari- ant tensors. Next, the nearly Kaehlerian (G;M)-structures under conformal change in a Rizza manifold are investigated.

Spin networks for noncompact groups

Spin networks are a natural generalization of Wilson loop functionals. They have been extensively studied in the case where the gauge group is compact and it has been shown that they naturally form a basis of gauge invariant observables. Physically the restriction to compact gauge groups is enough for the study of Yang–Mills theories, however it is well known that noncompact groups naturally arise as internal gauge groups for Lorentzian gravity models. In this context, a proper construction of gauge invariant observables is needed. The purpose of the present work is to define the notion of spin network states for noncompact groups. We first build, by a careful gauge fixing procedure, a natural measure and a Hilbert space structure on the space of gauge invariant graph connections. Spin networks are then defined as generalized eigenvectors of a complete set of hermitic commuting operators. We show how the delicate issue of taking the quotient of a space by noncompact groups can be address in term of algebraic geometry. We finally construct the full Hilbert space containing all spin network states. Having in mind applications to gravity, we illustrate our results for the groups SL(2,R) and SL(2,C).

A decomposition theorem for singular control systems on lie groups

In this paper, we introduce the notion of a singular control system S G on a connected finite-dimensional Lie group G with Lie algebra g . This definition depends on a pair of derivations ( E,D) of g where E plays the same roll as the singular matrix defining S R n and D induces the drift vector field of the system. Associated to E we construct a principal fibre bundle and an invariant connection which allow to us to obtain a decomposition result for S G via two subsystems: a linear control system and a differential-algebraic control system. We give an example on the simply connected Heisenberg Lie group of dimension three.

A formal moduli space of symplectic connections of Ricci-type on T2 n

We consider analytic curves ∇ t of symplectic connections of Ricci-type on the torus T 2 n with ∇ 0 the standard connection. We show, by a recursion argument, that if ∇ t is a formal curve of such connections then there exists a formal curve of symplectomorphisms ψ t such that ψ t ·∇ t is a formal curve of flat T 2 n -invariant symplectic connections and so ∇ t is flat for all t. Applying this result to the Taylor series of the analytic curve, it means that analytic curves of symplectic connections of Ricci-type starting at ∇ 0 are also flat. The group G of symplectomorphisms of the torus ( T 2 n , ω) acts on the space E of symplectic connections which are of Ricci-type. As a preliminary to study the moduli space E/G we study the moduli of formal curves of connections under the action of formal curves of symplectomorphisms.

CERTAIN CONFORMALLY INVARIANT CONNECTIONS OF RIZZA MANIFOLDS

We introduce certain conformally invariant h-Finsler connection in a Rizza manifold. Using this connection, we find some conformally invariant Finsler tensors. The conformal flatness and the Kaehlerian Finsler manifold with respect to the above connection are investigated.

Differentiable Connections with Poincar{e} Groups of Local Transformations. {III}

In the canonical smooth fiber bundles Π:∝n+1↑∝n, we study generalized differentiable connections constructed by the author in papers [4] and [5]. Special emphasis is laid on the investigation of the behavior of these connections under local transformations of the classical Poincar{e} groups ℙ(1,n) and extended Poincar{e} groups \(\tilde {\mathbb {P}}(1,n)\) canonically acting in the given connections. We found all first-order nonholonomic affine, Γ1, Γ2, and Γ1,2-connections with the groups ℙ(1,n) and \(\tilde {\mathbb {P}}(1,n)\) of local transformations and also constructed classes of the corresponding invariant second-order connections.

Construction of examples of b-completion

We survey some techniques for constructing b-completions. First, we give general results relating the b-completion of a product to the b-completion of its factors, the b-completion of a quotient, and the dependence of the b-completion on the bundle chosen to form it. Second, we relate (when possible) some b-completions to those of Riemannian metrics, and study invariant connections on homogenous spaces. Some particular examples are also considered.

Invariant linear connections on homogeneous symplectic varieties

We find all homogeneous symplectic varieties of connected semisimple Lie groups that admit an invariant linear connection.