Holonomy Algebra Research Articles

Lorentzian Symmetric Spaces Which are Einstein–Yang–Mills with Respect to Invariant Metric Connections

We classify four-dimensional connected simply-connected Lorentzian symmetric spaces M with connected nontrivial isotropy group furnishing solutions of the Einstein–Yang–Mills equations. Those solutions are built with respect to some invariant metric connection $$\Lambda $$ in the bundle of orthonormal frames of M and some diagonal metric on the holonomy algebra corresponding to $$\Lambda $$ .

Conformal deformations of a specific class of lorentzian manifolds with non-irreducible holonomy representation

Concerning holonomy theory or in the context of the existence of parallel spinors, Lorentzian manifolds with indecomposable, but non-irreducible holonomy representation have considerable significance. In this paper, we have comprehensively concentrated on conformal deformations of a particular class of four dimensional Lorentzian manifolds with indecomposable, non-irreducible holonomy representation which admit a recurrent light-like vector field. This type of Lorentzian manifolds are denoted by pr-waves and their holonomy algebra is contained in the parabolic algebra $ \big(\mathbb{R}\oplus \mbox{so(2)}\big)\ltimes \mathbb{R}^2 $. Moreover, it is mainly illustrated that for an arbitrary conformal diffeomorphism by inducing some specific structural conditions a pr-wave manifold behaves totally analogous to Einstein manifolds. Particularly, it is demonstrated that in some special circumstances the structure of a pr-wave manifold is precisely the same as a manifold equipped with a warped product metric.

English

Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two G-invariant metrics of arbitrary signature on homogenous space G/H are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, G-invariant metrics on homogenous space G/H implies that their holonomy algebra cannot be full. We give an algorithm for finding all left invariant metrics geodesically equivalent to a given left invariant metric on a Lie group. Using that algorithm we prove that no two left invariant metrics of any signature on sphere S3 are geodesically equivalent. However, we present examples of Lie groups that admit geodesically equivalent, nonproportional, left-invariant metrics.

Holonomy Classification of Lorentz-Kähler Manifolds

The classification problem for holonomy of pseudo-Riemannian manifolds is actual and open. In the present paper, holonomy algebras of Lorentz-K\"ahler manifolds are classified. A simple construction of a metric for each holonomy algebra is given. Complex Walker coordinates are introduced and described using the potential. Complex pp-waves are characterized in terms of the curvature, holonomy and the potential. Classification of Lorentz-K\"ahler symmetric spaces is reviewed.

On a Holonomy Flag of Non-holonomic Distributions

We give definition of a holonomy flag in subRiemannian geometry, a generalization of the Riemannian holonomy algebra, and calculate it for the 3D subRiemannian Lie groups for different connections. We rewrite and give new interpretation for the Codazzi equations for the (2,3)-distributions on SU(2) and the Heisenberg group. We calculate holonomy flag for Tanaka-Webster, Tanno, and Wagner connections.

The twofold way of super holonomy

Abstract There are two different notions of holonomy in supergeometry, the supergroup introduced by Galaev and our functorial approach motivated by super Wilson loops. Either theory comes with its own version of invariance of vectors and subspaces under holonomy. By our first main result, the Twofold Theorem, these definitions are equivalent. Our proof is based on the Comparison Theorem, our second main result, which characterises Galaev’s holonomy algebra as an algebra of coefficients, building on previous results. As an application, we generalise some of Galaev’s results to S-points, utilising the holonomy functor. We obtain, in particular, a de Rham–Wu decomposition theorem for semi-Riemannian S-supermanifolds.

The classification of naturally reductive homogeneous spaces in dimensions [formula omitted

We present a new method for classifying naturally reductive homogeneous spaces – i.e. homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion and curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel skew torsion on Riemannian manifolds and the interplay of such a connection with the geometric structure on the given Riemannian manifold. It allows to reproduce by easier arguments the known classifications in dimensions 3,4, and 5, and yields as a new result the classification in dimension 6. In each dimension, one obtains a ‘hierarchy’ of degeneracy for the torsion form, which we then treat case by case. For the completely degenerate cases, we obtain results that are independent of the dimension. In some situations, we are able to prove that any Riemannian manifold with parallel skew torsion has to be naturally reductive. We show that a ‘generic’ parallel torsion form defines a quasi-Sasakian structure in dimension 5 and an almost complex structure in dimension 6.

On the algebra of parallel endomorphisms of a pseudo-Riemannian metric

On a (pseudo-)Riemannian manifold (M,g), some fields of endomorphisms i.e. sections of End(TM) may be parallel for g. They form an associative algebra A, which is also the commutant of the holonomy group of g. As any associative algebra, A is the sum of its radical and of a semi-simple algebra S. We show in arXiv:1402.6642 that S may be of eight different types, including the generic type S=R.Id, and the K\"ahler and hyperk\"ahler types where S is respectively isomorphic to the complex field C or to the quaternions H. We show here that for any self adjoint nilpotent element N of the commutant of such an S in End(TM), the set of germs of metrics such that A contains S and {N} is non-empty. We parametrise it. Generically, the holonomy algebra of those metrics is the full commutant of $S\cup\{N\}$ in O(g). Apart from some "degenerate" cases, the algebra A is then $S \oplus (N)$, where (N) is the ideal spanned by N. To prove it, we introduce an analogy with complex Differential Calculus, the ring R[X]/(X^n) replacing the field C. This describes totally the local situation when the radical of A is principal and consists of self adjoint elements. We add a glimpse on the case where this radical is not principal.

How to Find the Holonomy Algebra of a Lorentzian Manifold

Manifolds with exceptional holonomy play an important role in string theory, supergravity and M-theory. It is explained how one can find the holonomy algebra of an arbitrary Riemannian or Lorentzian manifold. Using the de Rham and Wu decompositions, this problem is reduced to the case of locally indecomposable manifolds. In the case of locally indecomposable Riemannian manifolds, it is known that the holonomy algebra can be found from the analysis of special geometric structures on the manifold. If the holonomy algebra $${\mathfrak{g}\subset\mathfrak{so}(1,n-1)}$$ of a locally indecomposable Lorentzian manifold (M, g) of dimension n is different from $${\mathfrak{so}(1,n-1)}$$ , then it is contained in the similitude algebra $${\mathfrak{sim}(n-2)}$$ . There are four types of such holonomy algebras. Criterion to find the type of $${\mathfrak{g}}$$ is given, and special geometric structures corresponding to each type are described. To each $${\mathfrak{g}}$$ there is a canonically associated subalgebra $${\mathfrak{h} \subset\mathfrak{so}(n-2)}$$ . An algorithm to find $${\mathfrak{h}}$$ is provided.

Conformal holonomy in MacDowell-Mansouri gravity

The MacDowell-Mansouri formulation of general relativity is based on a gauge theory whose gauge algebra depends on the sign of the cosmological constant. In this article, we show that the gauge algebra is uniquely determined by the conformal structure of spacetime itself. Specifically, we show that in vacuum: the spacetime conformal holonomy algebra coincides with the MacDowell-Mansouri gauge algebra for both signs of the cosmological constant, in both Lorentzian and Euclidean metric signatures.

Systems of symplectic forms on four-manifolds

We study almost-Hermitian 4-manifolds with holonomy algebra, for the canonical Hermitian connection, of dimension at most one. We show how Riemannian 4-manifolds admitting five orthonormal symplectic forms fit in this picture and we classify them. In this set-up we also fully describe almost-K¨ahler 4-manifolds.

Note on the holonomy groups of pseudo-Riemannian manifolds

For an arbitrary subalgebra $\mathfrak{h}\subset\mathfrak{so}(r,s)$, a polynomial pseudo-Riemannian metric of signature $(r+2,s+2)$ is constructed, the holonomy algebra of this metric contains $\mathfrak{h}$ as a subalgebra. This result shows the essential distinction of the holonomy algebras of pseudo-Riemannian manifolds of index bigger or equal to 2 from the holonomy algebras of Riemannian and Lorentzian manifolds.

Holonomy algebras of pseudo-hyper-Kählerian manifolds of index 4

The holonomy algebra of a pseudo-hyper-Kählerian manifold of signature (4,4n+4) is a subalgebra of sp(1,n+1). Possible holonomy algebras of these manifolds are classified. Using this, a new proof of the classification of simply connected pseudo-hyper-Kählerian symmetric spaces of index 4 is obtained.

Riemannian manifolds with structure group PSU(3)

We study 8-dimensional Riemannian manifolds that admit a PSU(3)-structure. We classify these structures by their intrinsic torsion and characterize the corresponding classes via differential equations. Moreover, we consider a connection defined by a 3- and a 4-form that preserves the underlying structure. Finally, we discuss the geometry of these manifolds relatively to the holonomy algebra of this connection.

Prolongation on contact manifolds

On contact manifolds we describe a notion of (contact) finite-type for linear partial differential operators satisfying a natural condition on their leading terms. A large class of linear differential operators are of finite-type in this sense but are not well understood by currently available techniques. We resolve this in the following sense. For any such D we construct a partial connection r H on a (finite rank) vector bundle with the property that sections in the null space of D correspond bijectively, and via an explicit map, with sections parallel for the partial connection. It follows that the solution space of D is finite dimensional and bounded by the corank of the holonomy algebra of r H. The treatment is via a uniform procedure, even though in most cases no normal Cartan connection is available. The prolongations of a k th order linear differential operator between vector bundles arise by differentiating the given operator D : E ! F, and forming a new system comprising D along with auxiliary operators that capture some of this derived data. To exploit this effectively it is crucial to determine what part of this information should be retained, and then how best to manage it. With this understood, for many classes of operators the resulting prolonged operator can expose key properties of the original differential operator and its equation. Motivated by questions related to integrability and deformations of structure, a theory of overdetermined equations and prolonged systems was developed during the 1950s and 1960s by Goldschmidt, Spencer, and others (2, 17). Generally, results in these works are derived abstractly using jet bundle theory, and are severely restricted in the sense that they apply most readily to differential operators satisfying involutivity conditions. These features mean the theory can be difficult to apply. In the case that the given partial differential operator D : E ! F, has surjective symbol there is an effective algorithmic approach to this problem. The prolongations are constructed from the leading symbol �(D) : J k � 1 E ! F, where J k � 1 is the bundle of symmetric covariant tensors on M of rank k. At a point of M, denoting by K the kernel of �(D), the spaces K ` = ( J ` � 1 K) ( J k+` � 1 E), `� 0, capture spaces of new variables to be introduced, and the system closes up if K ` = 0 for

One component of the curvature tensor of a Lorentzian manifold

The holonomy algebra g of an n + 2 -dimensional Lorentzian manifold ( M , g ) admitting a parallel distribution of isotropic lines is contained in the subalgebra s i m ( n ) = ( R ⊕ s o ( n ) ) ⋉ R n ⊂ s o ( 1 , n + 1 ) . An important invariant of g is its s o ( n ) -projection h ⊂ s o ( n ) , which is a Riemannian holonomy algebra. One component of the curvature tensor of the manifold belongs to the space P ( h ) consisting of linear maps from R n to h satisfying an identity similar to the Bianchi one. In the present paper the spaces P ( h ) are computed for each possible h . This gives the complete description of the values of the curvature tensor of the manifold ( M , g ) . These results can be applied e.g. to the holonomy classification of the Einstein Lorentzian manifolds.

Holonomy of Einstein Lorentzian manifolds**Dedicated to Mark Volfovich Losik on his 75th birthday

The classification of all possible holonomy algebras of Einstein and vacuum Einstein Lorentzian manifolds is obtained. It is shown that each such algebra appears as the holonomy algebra of an Einstein (resp. vacuum Einstein) Lorentzian manifold; the direct constructions are given. Also the holonomy algebras of totally Ricci-isotropic Lorentzian manifolds are classified. The classification of the holonomy algebras of Lorentzian manifolds is reviewed and a complete description of the spaces of curvature tensors for these holonomies is given.

Cones over pseudo-Riemannian manifolds and their holonomy

By a classical theorem of Gallot (Ann. Sci. Éc. Norm. Sup. 12: 235–267, 1979), a Riemannian cone over a complete Riemannian manifold is either flat or has irreducible holonomy. We consider metric cones with reducible holonomy over pseudo-Riemannian manifolds. First we describe the local structure of the base of the cone when the holonomy of the cone is decomposable. For instance, we find that the holonomy algebra of the base is always the full pseudo-orthogonal Lie algebra. One of the global results is that a cone over a compact and complete pseudo-Riemannian manifold is either flat or has indecomposable holonomy. Then we analyse the case when the cone has indecomposable but reducible holonomy, which means that it admits a parallel isotropic distribution. This analysis is carried out, first in the case where the cone admits two complementary distributions and, second for Lorentzian cones. We show that the first case occurs precisely when the local geometry of the base manifold is para-Sasakian and that of the cone is para-Kählerian. For Lorentzian cones we get a complete description of the possible (local) holonomy algebras in terms of the metric of the base manifold.

Conformal holonomy of bi-invariant metrics

We discuss in this paper the conformal geometry of bi-invariant metrics on compact semisimple Lie groups. For this purpose, we develop an invariant Cartan calculus. Our main goal is to derive an iterative formula for the holonomy algebra of the normal conformal Cartan connection of a bi-invariant metric. As an example, we demonstrate the application of our invariant calculus to the computation of the conformal holonomy of SO(4). Its conformal holonomy algebra is so(7).

Projective holonomy II: cones and complete classifications

The aim of this paper and its prequel is to introduce and classify the irreducible holonomy algebras of the projective Tractor connection. This is achieved through the construction of a ‘projective cone’, a Ricci-flat manifold one dimension higher whose affine holonomy is equal to the Tractor holonomy of the underlying manifold. This paper uses the result to enable the construction of manifolds with each possible holonomy algebra.