Torsion-free Affine Connection Research Articles

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.

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.

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 $\mathbb{P}(TM)$.

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.

Complex affine distributions

Geometry of affine immersions is the study of hypersurfaces that are invariant under affine transformations. As with the hypersurface theory on the Euclidean space, an affine immersion can induce a torsion-free affine connection and a (pseudo)-Riemannian metric on the hypersurface. Moreover, an affine immersion can induce a statistical manifold, which plays a central role in information geometry. Recently, a statistical manifold with a complex structure is actively studied since it connects information geometry and Kähler geometry. However, a holomorphic complex affine immersion cannot induce such a statistical manifold with a Kähler structure. In this paper, we introduce complex affine distributions, which are non-integrable generalizations of complex affine immersions. We then present the fundamental theorem for a complex affine distribution, and show that a complex affine distribution can induce a statistical manifold with a Kähler structure.

Поля геометрических объектов, ассоциированных со скомпонованным гиперплоскостным H(Λ,L)-распреде­ле­ни­ем аффинного пространства

A compiled hyperplane distribution is considered in an n-dimensional projective space . We will briefly call it a -distribution. Note that the plane L(A) is the distribution characteristic obtained by displacement in the center belonging to the L-subbundle. The following results were obtained: a) The existence theorem is proved: -distribution exists with arbitrary (3n – 5) functions of n arguments. b) A focal manifold is constructed in the normal plane of the 1st kind of L-subbundle. It was obtained by shifting the cen­ter A along the curves belonging to the L-distribution. A focal manifold is also given, which is an analog of the Koenigs plane for the distribution pair (L, L). c) It is shown that a framed -distribution in the 1st kind normal field of H-distribution induces tangent and normal bundles. d) Six connection theorems induced by a framed -distri­bu­tion in these bundles are proved. In each of the bundles , the framed -distribution induces an intrin­sic torsion-free affine connection in the tangent bundle and a centro-affine connection in the corresponding normal bundle. e) In each of the bundles (d) in the differential neighborhood of the 2nd order, the covers of 2-forms of curvature and curvature tensors of the corresponding connections are constructed.

Conformal Patterson–Walker metrics

The classical Patterson-Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We achieve a complete description of Einstein metrics in the conformal class formed by the Patterson-Walker metric. Finally, we describe all symmetries of the conformal Patterson-Walker metric. In both cases we obtain descriptions in terms of geometric data on the original structure.

Metrics in Projective Differential Geometry: The Geometry of Solutions to the Metrizability Equation

Pseudo-Riemannian metrics with Levi-Civita connection in the projective class of a given torsion-free affine connection are equivalent to the maximal rank solutions of a certain overdetermined projectively invariant differential equation often called the metrizability equation. Dropping this rank assumption, we study the solutions to this equation, given less restrictive generic conditions on its prolonged system. In this setting, we find that the solution stratifies the manifold according to the strict signature (pointwise) of the solution and does this in way that locally generalizes the stratification of a model, where the model is, in each case, a corresponding Lie group orbit decomposition of the sphere. Thus the solutions give curved generalizations of such embedded orbit structures. We describe the smooth nature of the strata and determine the geometries of each of the different strata types; this includes a metric on the open strata that becomes singular at the strata boundary, with the latter a type of projective infinity for the given metric. The work also provides new results for the projective compactification of scalar-flat metrics.

Kenmotsu statistical manifolds and warped product

A notion of a Kenmotsu statistical manifold is introduced, which is locally obtained as the warped product of a holomorphic statistical manifold and a line. A statistical manifold is a Riemannian manifold equipped with a torsion-free affine connection satisfying the Codazzi equation. It can be considered as being in information geometry, Hessian geometry and various submanifold theory. On the other hand, a Kenmotsu manifold is in a meaningful class of almost contact metric manifolds. In this paper, we construct a suitable statistical structure on it. Although the notion of the warped product of Riemannian manifolds is well known, the one for statistical manifolds is not established. We consider it in general, and study the statistical sectional curvature of the warped product of two statistical manifolds. We show that a Kenmotsu statistical manifold of constant $$\phi $$ -sectional curvature is constructed from a special Kahler manifold, which is an important example of holomorphic statistical manifold. A Sasakian statistical manifold is also studied from the viewpoint of the warped product of statistical manifolds.

Transformations and coupling relations for affine connections

The statistical structure on a manifold M is predicated upon a special kind of coupling between the Riemannian metric g and a torsion-free affine connection ∇ on TM, such that ∇g is totally symmetric, forming, by definition, a “Codazzi pair” {∇,g}. In this paper, we first investigate various transformations of affine connections, including additive translation by an arbitrary (1,2)-tensor K, multiplicative perturbation through an arbitrary invertible operator L on TM, and conjugation through a non-degenerate bilinear form h. We then study the Codazzi coupling of ∇ with h and its coupling with L, and the link between these two couplings. We introduce, as special cases of K-translations, various transformations that generalize traditional projective and dual-projective transformations, and study their commutativity with L-perturbation and h-conjugation transformations. Our framework allows affine connections to carry torsion, and we investigate conditions under which torsions are preserved by the various transformations mentioned above. In addition to reproducing some known results regarding Codazzi transformations, conformal-projective transformations, etc., we extend many of these geometric relations, and hence obtain new geometric insights, for the general case of a non-degenerate bilinear form h (not required to be a symmetric form g) in relation to an affine connection with possibly non-vanishing torsion. In particular, we provide a generalization to the conformal-projective transformation of {∇,g} which preserves their Codazzi coupling. Our systematic approach establishes a general setting for the study of information geometry based on transformations and coupling relations between affine connections.

The Atiyah class of a dg-vector bundle

We introduce the notions of Atiyah class and Todd class of a differential graded vector bundle with respect to a differential graded Lie algebroid. We prove that the space of vector fields X(M) on a dg-manifold M with homological vector field Q admits a structure of L∞[1]-algebra with the Lie derivative LQ as unary bracket λ1, and the Atiyah cocycle AtM corresponding to a torsion-free affine connection as binary bracket λ2.

Pseudo-Riemannian Symmetries on Heisenberg Groups

Abstract The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on Z2-grading on Lie algebras. We consider homogeneous spaces G/H such that the Lie algebra g of G admits a Γ-grading where Γ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group H3 adapted to the symmetries of a Γ-symmetric structure on H3. We prove that the classification of Riemannian and Lorentzian Zl-symmetric metrics on H3 corresponds to the classification of its left-invariant Riemannian and Lorentzian metrics, up to isometry. We study also the Z§-symmetric structures on G/H when G is the (2p + 1)-dimensional Heisenberg group. This gives examples of non-Riemannian symmetric spaces. When k > 1, we show that there exists a family of flat and torsion free affine connections adapted to the Z§-symmetric structures.

Manifolds with holonomy $$U^*(2m)$$ U ∗ ( 2 m )

We consider the geometry determined by a torsion-free affine connection whose holonomy lies in the subgroup \(U^*(2m)\), a real form of \(GL(2m,\mathbf {C})\), otherwise denoted by \(SL(m,\mathbf {H}) \cdot U(1)\). We show in particular how examples may be generated from quaternionic Kahler or hyperkahler manifolds with a circle action.

Einstein metrics in projective geometry

It is well known that pseudo-Riemannian metrics in the projective class of a given torsion free affine connection can be obtained from (and are equivalent to) the solutions of a certain overdetermined projectively invariant differential equation. This equation is a special case of a so-called first BGG equation. The general theory of such equations singles out a subclass of so-called normal solutions. We prove that non-degerate normal solutions are equivalent to pseudo-Riemannian Einstein metrics in the projective class and observe that this connects to natural projective extensions of the Einstein condition.

Single Basepoint Subdivision Schemes for Manifold-valued Data: Time-Symmetry Without Space-Symmetry

This paper establishes smoothness results for a class of nonlinear subdivision schemes, known as the single basepoint manifold-valued subdivision schemes, which shows up in the construction of wavelet-like transform for manifold-valued data. This class includes the (single basepoint) Log–Exp subdivision scheme as a special case. In these schemes, the exponential map is replaced by a so-called retraction map f from the tangent bundle of a manifold to the manifold. It is known that any choice of retraction map yields a C 2 scheme, provided the underlying linear scheme is C 2 (this is called “C 2 equivalence”). But when the underlying linear scheme is C 3, Navayazdani and Yu have shown that to guarantee C 3 equivalence, a certain tensor P f associated to f must vanish. They also show that P f vanishes when the underlying manifold is a symmetric space and f is the exponential map. Their analysis is based on certain “C k proximity conditions” which are known to be sufficient for C k equivalence. In the present paper, a geometric interpretation of the tensor P f is given. Associated to the retraction map f is a torsion-free affine connection, which in turn defines an exponential map. The condition P f =0 is shown to be equivalent to the condition that f agrees with the exponential map of the connection up to the third order. In particular, when f is the exponential map of a connection, one recovers the original connection and P f vanishes. It then follows that the condition P f =0 is satisfied by a wider class of manifolds than was previously known. Under the additional assumption that the subdivision rule satisfies a time-symmetry, it is shown that the vanishing of P f implies that the C 4 proximity conditions hold, thus guaranteeing C 4 equivalence. Finally, the analysis in the paper shows that for k≥5, the C k proximity conditions imply vanishing curvature. This suggests that vanishing curvature of the connection associated to f is likely to be a necessary condition for C k equivalence for k≥5.

On Holomorphic Hypercomplex Connections

The main purpose of this paper is to study some properties of integrable commutative hypercomplex structures endowed with a holomorphic torsion-free affine connection whose curvature tensor satisfies the purity condition with respect to the covariantly constant structure affinors.

Moduli spaces of framed instanton bundles on [formula omitted] and twistor sections of moduli spaces of instantons on [formula omitted

We show that the moduli space M of framed instanton bundles on CP 3 is isomorphic (as a complex manifold) to a subvariety in the moduli of rational curves of the twistor space of the moduli space of framed instantons on R 4 . We then use this characterization to prove that M is equipped with a torsion-free affine connection with holonomy in Sp ( 2 n , C ) .

Affine connections, midpoint formation, and point reflection

We describe some differential-geometric structures in combinatorial terms: namely affine connections and their torsion and curvature, and we show that torsion free affine connections may equivalently be presented in terms of some simpler combinatorial structure: midpoint formation, and point reflection (geodesic symmetry). The method employed is that of synthetic differential geometry, which is briefly explained.

Structure of the curvature tensor on symplectic spinors

We study symplectic manifolds ( M 2 l , ω ) equipped with a symplectic torsion-free affine (also called Fedosov) connection ∇ and admitting a metaplectic structure. Let S be the so-called symplectic spinor bundle over M and let R S be the curvature field of the symplectic spinor covariant derivative ∇ S associated to the Fedosov connection ∇ . It is known that the space of symplectic spinor valued exterior differential 2 -forms, Γ ( M , ⋀ 2 T ∗ M ⊗ S ) , decomposes into three invariant subspaces with respect to the structure group, which is the metaplectic group M p ( 2 l , R ) in this case. For a symplectic spinor field ϕ ∈ Γ ( M , S ) , we compute explicitly the projections of R S ϕ ∈ Γ ( M , ⋀ 2 T ∗ M ⊗ S ) onto the three mentioned invariant subspaces in terms of the symplectic Ricci and symplectic Weyl curvature tensor fields of the connection ∇ . Using this decomposition, we derive a complex of first order differential operators provided the Weyl curvature tensor field of the Fedosov connection is trivial.