Torsion-free Connection Research Articles

Some computational results on Koszul–Vinberg cochain complexes

AbstractAn affine connection is said to be flat if its curvature tensor vanishes identically. Koszul–Vinberg (KV for abbreviation) cohomology has been invoked to study the deformation theory of flat and torsion-free affine connections on tangent bundle. In this Note, we compute explicitly the differentials of various specific KV cochains, and study their relation to classical objects in information geometry, including deformations associated with projective and dual-projective transformations of a flat and torsion-free affine connection. As an application, we also give a simple yet non-trivial example of a left-symmetric algebra of which the second cohomology group does not vanish.

Silver Structures on the Riemann Extensions

In the present paper we deal with an $n-$dimensional differentiable manifold $M$ with a torsion-free linear connection $\nabla $. Here we study some properties of a silver structure on the cotangent bundle ${{T}^{*}}M$ equipped with the Riemannian extension ${{}^{R}}\nabla$ and obtain a necessary condition for which the silver semi-Riemannian manifold $\left( {{T}^{*}}M{{,}^{R}}\nabla ,S \right)$ to be a locally decomposable.

Explicit maximal totally real embeddings

This article deals with an explicit canonical construction of a maximal totally real embedding for real analytic manifolds equipped with a covariant derivative operator acting on the real analytic sections of its tangent bundle or of its complexified tangent bundle. The existence of maximal totally real embeddings for real analytic manifolds is known from previous celebrated works by Bruhat-Whitney [1] and Grauert [4]. Their construction is based on the use of analytic continuation of local frames and local coordinates that are far from being canonical or explicit. As a consequence, the form of the corresponding complex structure has been a mystery since the very beginning. A quite simple recursive expression for such complex structures has been provided in the first author's work “On maximal totally real embeddings” [12]. In our series of articles we focus on the case of torsion free connections. In the present article we give a fiberwise Taylor expansion of the canonical complex structure which is expressed in terms of symmetrization of curvature monomials and a rather simple and explicit expression of the coefficients of the expansion. We explain also a rather simple geometric characterization of such canonical complex structures. Our main result and argument can be useful for the study of open questions in the theory of the embeddings in consideration such as their moduli space.

Characteristic conic connections and torsion-free principal connections

We study the relation between torsion tensors of principal connections on G-structures and characteristic conic connections on associated cone structures. We formulate sufficient conditions under which the existence of a characteristic conic connection implies the existence of a torsion-free principal connection. We verify these conditions for adjoint varieties of simple Lie algebras, excluding those of type Aℓ≠2 or Cℓ. As an application, we give a complete classification of the germs of minimal rational curves whose VMRT at a general point is such an adjoint variety: nontrivial ones come from lines on hyperplane sections of certain Grassmannians or minimal rational curves on wonderful group compactifications.

Metrizability of SO(3)-invariant connections: Riemann versus Finsler

Abstract For a torsion-free affine connection on a given manifold, which does not necessarily arise as the Levi-Civita connection of any pseudo-Riemannian metric, it is still possible that it corresponds in a canonical way to a Finsler structure; this property is known as Finsler (or Berwald-Finsler) metrizability.
In the present paper, we clarify, for 4-dimensional SO(3)-invariant, Berwald-Finsler metrizable connections, the issue of the existence of an affinely equivalent pseudo-Riemannian structure. In particular, we find all classes of SO (3)-invariant connections which are not Levi-Civita connections for any pseudo-Riemannian metric - hence, are non-metric in a conventional way - but can still be metrized by SO(3)-invariant Finsler functions. The implications for physics, together with some examples are briefly discussed.

On the cotangent bundle with vertical modified riemannian extensions

Let $M$ be an n-dimensional differentiable manifold with a torsion-free linear connection $\nabla $ which induces on its cotangent bundle ${T^*}M$. The main purpose of the present paper is to study some properties of the vertical modified Riemannian extension on ${T^*}M$ which is given as a new metric in [17]. At first, we investigate a metric connection with torsion on ${T^*}M$. And then, we present the holomorphy properties with respect to a compatible almost complex structure. urthermore, we study locally decomposable Golden pseudo-Riemannian structures on the cotangent bundle endowed with vertical modified Riemannian extension.

Noncommutative Riemannian geometry of Kronecker algebras

We study aspects of noncommutative Riemannian geometry of the path algebra arising from the Kronecker quiver with N arrows. To start with, the framework of derivation based differential calculi is recalled together with a discussion on metrics and bimodule connections compatible with the ⁎-structure of the algebra. As an illustration, these concepts are applied to the noncommutative torus where examples of torsion free and metric (Levi-Civita) connections are given.In the main part of the paper, noncommutative geometric aspects of (generalized) Kronecker algebras are considered. The structure of derivations and differential calculi is explored, and torsion free bimodule connections are studied together with their compatibility with hermitian forms, playing the role of metrics on the module of differential forms. Moreover, for several different choices of Lie algebras of derivations, non-trivial Levi-Civita connections are constructed.

Average structures associated to a Finsler space

Given a Finsler space (M, F ) on a manifold M , the averaging method associates to Finslerian geometric objects affine geometric objects living on M . In particular, a Riemannian metric is associated with the fundamental tensor g and an affine, torsion free connection is associated with the Chern-Rund connection. As an illustration of the applications of theory, a generalization of the Gauss-Bonnet theorem to Berwald surfaces using the average metric is presented. The parallel transport and curvature endomorphisms of the average connection are obtained. The holonomy group for a Berwald space is discussed. Finally, isometries and symmetric spaces are considered and the heritage of the property of symmetric space from the Finsler space to the average Riemannian metric is proved.

Аналоги симметрической и плоской связностей с нетензорами кручения и кривизны

The paper is devoted to affine connection in the frame bundle associ­ated with a manifold which structure equations and derivation formulas are constructed using deformations of the exterior and ordinary differen­tials. Curvature and torsion objects of this connection are not tensors. A cha­racteristic of a curvature which is a convolution of a deformation tensor and a torsion, is considered. Torsion-free connections are not distingui­shed on the introduced manifold, even in the case of symmetric deforma­tion, a class of semi-symmetric connections is distinguished, which is an analogue of symmetric connection on an ordinary smooth manifold. It is proved that if the connection deformation tensor is symmetric or zero, then the connection is semi-symmetric. Analogues of torsion-free and cur­vature-free connections are constructed. The torsion and curvature of this connection are expressed in terms of the symmetric deformation tensor for the connection. Canonical connection is a special case of this connec­tion, it is semi-symmetric and curvature-free.

Torsion-free connections on G-structures

We prove that for a Lie group SOn(R)⊂G⊂GLn(R), any G-structure on a smooth manifold can be endowed with a torsion free connection which is locally the Levi-Civita connection of a Riemannian metric in a given conformal class. In this process, we classify the admissible groups.

Gronwall's conjecture for 3-webs with two pencils of lines

We prove the old-standing Gronwall conjecture in the particular case of linear 3-webs whose 2 foliations are 2 pencils of lines. For a non-hexagonal 3-web, we also introduce a family of projective torsion-free Cartan connections, the web leaves being geodesics for each member of the family, and give a web linearization criterion.

Conelike radiant structures

Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulation of Einstein equations for special statistical structures that generalizes the usual Einstein equations for pseudo-Riemannian metrics and is of independent interest. A conelike radiant structure is a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure is a torsion-free affine connection and a vector field whose covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesic surface passing through the point and tangent to the subspace. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra. The affine connection of a conelike radiant structure can be normalized in a canonical way to have antisymmetric Ricci tensor. Applied to a conelike radiant structure on the total space of a principal bundle with one-dimensional fiber this yields a generalization of the classical Thomas connection of a projective structure. The compatibility of radiant and conelike structures with metrics is investigated and yields a construction of connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein–Weyl structures over Kähler–Einstein manifolds having nonzero scalar curvature. A formulation of Einstein equations for special statistical manifolds is given that generalizes the Einstein–Weyl equations and encompasses these more general examples. There are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail.

Counting holomorphic connections with a prescribed Ricci tensor

How many holomorphic connections are there with a prescribed Ricci tensor? How many torsion-free holomorphic connections are there with a prescribed Ricci tensor? These questions are answered by using the holomorphic version of the Cauchy–Kowalevski theorem.

Locally conformally Hessian and statistical manifolds

A statistical manifold (M,D,g) is a manifold M endowed with a torsion-free connection D and a Riemannian metric g such that the tensor Dg is totally symmetric. If D is flat then (M,g,D) is a Hessian manifold. A locally conformally Hessian (l.c.H.) manifold is a quotient of a Hessian manifold (C,∇,g) such that the monodromy group acts on C by Hessian homotheties, i.e. this action preserves ∇ and multiplies g by a group character. The l.c.H. rank is the rank of the image of this character considered as a function from the monodromy group to real numbers. A l.c.H. manifold is called radiant if the Lee vector field ξ is Killing and satisfies ∇ξ=λId. We prove that the set of radiant l.c.H. metrics of l.c.H. rank 1 is dense in the set of all radiant l.c.H. metrics. We prove a structure theorem for compact radiant l.c.H. manifold of l.c.H. rank 1. Every such manifold C is fibered over a circle, the fibers are statistical manifolds of constant curvature, the fibration is locally trivial, and C is reconstructed from the statistical structure on the fibers and the monodromy automorphism induced by this fibration.

Notes on some properties of the natural Riemann extension

Let $(M,\nabla)$ be an $n$-dimensional differentiable manifold with a torsion-free linear connection and $T^{*}M$ its cotangent bundle. In this context we study some properties of the natural Riemann extension (M. Sekizawa (1987), O. Kowalski and M. Sekizawa (2011)) on the cotangent bundle $T^{*}M$. First, we give an alternative definition of the natural Riemann extension with respect to horizontal and vertical lifts. Secondly, we investigate metric connection for the natural Riemann extension. Finally, we present geodesics on the cotangent bundle $T^{*}M$ endowed with the natural Riemann extension.

Null-Projectability of Levi-Civita Connections

We study the natural property of projectability of a torsion-free connection along a foliation on the underlying manifold, which leads to a projected torsion-free connection on a local leaf space, focusing on projectability of Levi-Civita connections of pseudo-Riemannian metrics along foliations tangent to null parallel distributions. For the neutral metric signature and mid-dimensional distributions, Afifi showed in 1954 that projectability of the Levi-Civita connection characterizes, locally, the case of Patterson and Walker’s Riemann extension metrics. We extend this correspondence to null parallel distributions of any dimension, introducing a suitable generalization of Riemann extensions.

Probing geometric proca in metric-palatini gravity with black hole shadow and photon motion

Extended metric-Palatini gravity, quadratic in the antisymmetric part of the affine curvature, is known to lead to the general relativity plus a geometric Proca field. The geometric Proca, equivalent of the non-metricity vector in the torsion-free affine connection, qualifies to be a distinctive signature of the affine curvature. In the present work, we explore how shadow and photon motion near black holes can be used to probe the geometric Proca field. To this end, we derive static spherically symmetric field equations of this Einstein-geometric Proca theory, and show that it admits black hole solutions in asymptotically AdS background. We perform a detailed study of the optical properties and shadow of this black hole and contrast them with the observational data by considering black hole environments with and without plasma. As a useful astrophysical application, we discuss constraints on the Proca field parameters using the observed angular size of the shadow of supermassive black holes M87^* and Sgr A^* in both vacuum and plasma cases. Overall, we find that the geometric Proca can be probed via the black hole observations.

Affine connections on complex compact surfaces and Riccati distributions

Let $M$ be a complex surface. We show that there is a one-to-one correspondence between torsion-free affine connections on $M$ and Riccati distributions on $\mathbb{P}(TM)$. Furthermore, if $M$ is compact, then this correspondence induces a one-to-one correspondence between affine structures on $M$ and Riccati foliations on $\mathbb{P}(TM)$.

Lorentzian non-stationary dynamical systems

In this paper, we introduce a Lorentzian Anosov family (LA-family) up to a sequence of distributions of null vectors. We prove for each p ∈ Mi , where Mi is a Lorentzian manifold for i ∈ ℤ the tangent space Mi at p has a unique splitting and this splitting varies continuously on a sequence via the distance function created by a unique torsion-free semi-Riemannian connection. We present three examples of LA-families. Also, we define Lorentzian shadowing property of type I and II and prove some results related to this property.

Minimal Rational Curves and 1-Flat Irreducible G-Structures

1-Flat irreducible G-structures, equivalently, irreducible G-structures admitting torsion-free affine connections, have been studied extensively in differential geometry, especially in connection with the theory of affine holonomy groups. We propose to study them in a setting in algebraic geometry, where they arise from varieties of minimal rational tangents (VMRT) associated to families of minimal rational curves on uniruled projective manifolds. We prove that such a structure is locally symmetric when the dimension of the uniruled projective manifold is at least 5. By the classification result of Merkulov and Schwachhöfer on irreducible affine holonomy, the problem is reduced to the case when the VMRT at a general point of the uniruled projective manifold is isomorphic to a subadjoint variety. In the latter situation, we prove a stronger result that, without the assumption of 1-flatness, the structure arising from VMRT is always locally flat. The proof employs the method of Cartan connections. An interesting feature is that Cartan connections are considered not for the G-structures themselves, but for certain geometric structures on the spaces of minimal rational curves.