Torsion-free Affine Connection Research Articles

Complex of twistor operators in symplectic spin geometry

For a symplectic manifold admitting a metaplectic structure (a symplectic analogue of the Riemannian spin structure), we construct a sequence consisting of differential operators using a symplectic torsion-free affine connection. All but one of these operators are of first order. The first order ones are symplectic analogues of the twistor operators known from Riemannian spin geometry. We prove that under the condition the symplectic Weyl curvature tensor field of the symplectic connection vanishes, the mentioned sequence forms a complex. This gives rise to a new complex for the so called Ricci type symplectic manifolds, which admit a metaplectic structure.

Ricci curvature of affine connections

We show various examples of torsion-free affine connections which preserve volume elements and have definite Ricci curvature tensors.

Ricci-flat holonomy: A classification

The reductive holonomy algebras for a torsion-free affine connection are analysed, with the goal of establishing which ones can correspond to a Ricci-flat connection with the same properties. Various families of holonomies are eliminated through different algebraic means, and examples are constructed (in this paper and in ‘Projective Holonomy II: Cones and Complete Classification’, by the same author) in the remaining cases, thus solving this problem completely, for reductive holonomy.

Banach manifolds of algebraic elements in the algebra $$\mathcal{L}$$ (H) of bounded linear operatorsof bounded linear operators

Given a complex Hilbert space H, we study the manifold\(\mathcal{A}\) of algebraic elements in\(Z = \mathcal{L}\left( H \right)\). We represent\(\mathcal{A}\) as a disjoint union of closed connected subsets M of Z each of which is an orbit under the action of G, the group of all C*-algebra automorphisms of Z. Those orbits M consisting of hermitian algebraic elements with a fixed finite rank r, (0< r<∞) are real-analytic direct submanifolds of Z. Using the C*-algebra structure of Z, a Banach-manifold structure and a G-invariant torsionfree affine connection ∇ are defined on M, and the geodesics are computed. If M is the orbit of a finite rank projection, then a G-invariant Riemann structure is defined with respect to which ∇ is the Levi-Civita connection.

Affine hypersurfaces with the Ricci semi-symmetry

The purpose of the present paper is to prove that either a proper affine hypersphere or an affine cylindrical (See Introduction) is the only nondegenerate affine hypersurface of affine space with torsion free affine connection which satisfies the Ricci semi-symmetry and to study the equivalence of semi-symmetric and Ricci semi-symmetric in the case of affine hypersurfaces

Characterization of Gradient Control Systems

Given a general nonlinear affine control system with outputs and a torsion-free affine connection defined on its state space, we investigate the gradient realization problem: we give necessary and sufficient conditions under which the control system can be written as a gradient control system corresponding to some pseudo-Riemannian metric whose Levi-Civita connection is equal to the given affine connection. The results rely on a suitable notion of compatibility of the system with respect to the given affine connection, and on the output behavior of the prolonged system and the gradient extension. The symmetric product associated with an affine connection plays a key role throughout the discussion.

Generalized Relative Entropies as Contrast Functionals on Density Matrices

We use a class of generalized relative entropies on density matrices to obtain one-parameter families of torsion-free affine connections.

The Chern Conjecture for Affinely Flat Manifolds Using Combinatorial Methods

An affine manifold is a manifold with a flat affine structure, i.e. a torsion-free flat affine connection. We slightly generalize the result of Hirsch and Thurston that if the holonomy of a closed affine manifold is isomorphic to amenable groups amalgamated or HNN-extended along finite groups, then the Euler characteristic of the manifold is zero confirming an old conjecture of Chern. The technique is from Kim and Lee's work using the combinatorial Gauss–Bonnet theorem and taking the means of the angles by amenability. We show that if an even-dimensional manifold is obtained from a connected sum operation from K(π, 1)s with amenable fundamental groups, then the manifold does not admit an affine structure generalizing a result of Smillie.

Zoll Manifolds and Complex Surfaces

We classify compact surfaces with torsion-free affine connections for which every geodesic is a simple closed curve. In the process, we obtain completely new proofs of all the major results [4] concerning the Riemannian case. In contrast to previous work, our approach is twistor-theoretic, and depends fundamentally on the fact that, up to biholomorphism, there is only one complex structure on ℂℙ2.

The manifold of finite rank projections in the algebra ℒ(H) of bounded linear operators

Given a complex Hilbert space H, we study the differential geometry of the manifold M of all projections in V = ℒ( H). Using the algebraic structure of V, a torsionfree affine connection ∇ (that is invariant under the group of automorphisms of V) is defined on every connected component m of M, which in this way becomes a symmetric holomorphic manifold that consists of projections of the same rank r, (0 ≤ r ≤ ∞). We prove that m admits a Riemann structure if and only if m consists of projections that have the same finite rank r or the same finite corank, and in that case ∇ is the Levi-Civita and the Kähler connection of m. Moreover, m turns out to be a totally geodesic Riemann manifold whose geodesics and Riemann distance are computed.

Geometry of quantum states: dual connections and divergence functions

In a finite quantum state space with a monotone metric, a family of torsion-free affine connections is introduced, in analogy with the classical α-connections defined by Amari. The dual connections with respect to the metric are found and it is shown that they are, in general, not torsion-free. The torsion and the Riemannian curvature are computed and the existence of efficient estimators is treated. Finally, geodesics are used to define a divergence function.

Classification of Irreducible Holonomies of Torsion-Free Affine Connections

The subgroups of GL(n,R) that act irreducibly on R^n and that can occur as the holonomy of a torsion-free affine connection on an n-manifold are classified, thus completing the work on this subject begun by M. Berger in the 1950s. The methods employed include representation theory, the theory of hermitian symmetric spaces, twistor theory, and Poisson geometry. The latter theory is especially important for the construction and classification of those torsion-free connections whose holonomy falls into one of the so-called `exotic' cases, i.e., those that were not included in Berger's original lists. Some remarks involving an interpretation of some of the examples in terms of supersymmetric constructions are also included.

Compact Einstein-Weyl manifolds and the associated constant

A Weyl structure on a manifold M is a torsion free affine connection D preserving a conformal structure [g]. Namely, a torsion free affine connection D is called a Weyl structure if Dg — ω ® g for a 1-form ω. The definition of Weyl structure goes back to the work of H. Weyl. In his famous book ([23]) he introduced Weyl structure to unify gravitational fields and electromagnetic fields. The notion of Einstein-Weyl structure is originated in the paper of N.Hitchin ([11]) in which he developed the 3-dimensional minitwistor theory associated to the 3-dimensional monopole theory and observed that the minitwistor theory can be generalized over a 3-manifold endowed with a Weyl structure obeying a certain Ricci tensor condition, namely an Einstein-Weyl structure. Refer also to [12]. The exact definition of Einstein-Weyl structure is the following. A Weyl structure (-D, [g]) is Einstein-Weyl if the symmetrized Ricci tensor is proportional to a metric g representing [#];

Exotic Holonomies ${\rm E}^{(a)}_7$

It is proved that the Lie groups [Formula: see text] and [Formula: see text] represented in ℝ56 and the Lie group [Formula: see text] represented in ℝ112 occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension.

On dual holomorphically projectively flat affine connections

Given a complex Riemannian metrich and a torsion-free complex affine connection ▽ on a complex manifold, a dual holomorphically-planar curve of ▽ is defined as a curve whose tangent complex plane, generated by its tangent 1-form, is parallel along the curve. The corresponding dual holomorphically projective group is defined as a group of transformations of connections preserving dual holomorphically-planar curves. The class of connections complex semi-compatible with the metrich and pairs of complex semi-conjugate connections are defined using the relations between their holomorphically-planar curves and their dual holomorphically-planar curves. The dual holomorphically-projective curvature tensor for a connection complex semi-compatible withh is determined as an invariant of the dual holomorphically-projective group. Dual holomorphically-projectively flat connections complex semi-compatible withh are characterized as connections with vanishing dual holomorphically-projective curvature tensor.

On Affine Connections whose Holonomy is a Tensor Representation

In 1955, Berger [4] gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. While this list was stated to be complete in the case of metric connections, the situation in the general case remained unclear. The (non-metric) representations which are missing from this list are called exotic. In recent years, it has been determined that exotic holonomies do exist. Thus, Berger’s classification is yet to be completed in the non-metric case. In this paper, we investigate certain holonomy representations of reductive Lie groups whose semi-simple part is not simple.

On the existence of infinite series of exotic holonomies

In 1955, Berger \cite{Ber} gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. This list was stated to be complete up to possibly a finite number of missing entries. In this paper, we show that there is, in fact, an infinite family of representations which are missing from this list, thereby showing the incompleteness of Berger's classification. Moreover, we develop a method to construct torsion-free connections with prescribed holonomy, and use it to give a complete description of the torsion-free affine connections with these new holonomies. We also deduce some striking facts about their global behaviour.

Left-Invariant Affine Structures on Reductive Lie Groups

We describe left-invariant affine structures (that is, left-invariant flat torsion-free affine connections ∇) on reductive linear Lie groupsG. They correspond bijectively to LSA-structures on the Lie algebra g ofG. Here LSA stands for left-symmetric algebra. If g has trivial or one-dimensional center z then the affine representation α=λ⊕1 of g, induced by any LSA-structure gλon g isradiant, i.e., the radiance obstructioncα∈H1(g,gλ) vanishes. If dimz=1 we prove that g=s⊕z, where s is split simple, admits LSA-structures if and only if s is of typeAl, that is, g=gln.Here we have the associative LSA-structure given by ordinary matrix multiplication corresponding to the bi-invariant affine structure on GL(n), which was believed to be essentially the only possible LSA-structure on gln. We exhibit interesting LSA-structures different from the associative one. They arise as certain deformations of the matrix algebra. Then we classify all LSA-structures on glnusing a result of Baues. Forn=2 we compute all structures explicitly over the complex numbers.

Geometry of Kodaira moduli spaces

A general theorem on the existence of natural torsion-free affine connections on a complete family of compact complex submanifolds in a complex manifold is proved.

On dual-projectively flat affine connections

Given a pseudo Riemannian metrich and a torsion-free affine connection ∇ on a smoothn-manifold M,a dual geodesic curve of ∇ is defined as a curve whose tangent 1-form is parallel along the curve. The corresponding dual-projective group is defined as a group of transformations of connections preserving dual-geodesic curves. The class of connections semi-compatible with the metrich and pairs of semi-conjugate connections are defined using the relations between their geodesics and dual-geodesics. The dual-projective curvature tensor for a connection semi-compatible withh is determined as an invariant of the dual projective group. Dual-projectively flat connections semi-compatible withh are characterized as connections with vanishing dual-projective curvature tensor. As an application we recover the fundamental theorem for non-degenerate hypersurface immersions.