Projective Differential Geometry Research Articles

Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients

We show that 3-Sasaki structures admit a natural description in terms of projective differential geometry. First we establish that a 3-Sasaki structure may be understood as a projective structure whose tractor connection admits a holonomy reduction, satisfying a particular non-vanishing condition, to the (possibly indefinite) unitary quaternionic group Sp(p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Sp}}(p,q)$$\\end{document}. Moreover, we show that, if a holonomy reduction to Sp(p,q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ ext {Sp}}(p,q)$$\\end{document} of the tractor connection of a projective structure does not satisfy this condition, then it decomposes the underlying manifold into a disjoint union of strata including open manifolds with (indefinite) 3-Sasaki structures and a closed separating hypersurface at infinity with respect to the 3-Sasaki metrics. It is shown that the latter hypersurface inherits a Biquard–Fefferman conformal structure, which thus (locally) fibers over a quaternionic contact structure, and which in turn compactifies the natural quaternionic Kähler quotients of the 3-Sasaki structures on the open manifolds. As an application, we describe the projective compactification of (suitably) complete, non-compact (indefinite) 3-Sasaki manifolds and recover Biquard’s notion of asymptotically hyperbolic quaternionic Kähler metrics.

К ПРОЕКТИВНО–ДИФФЕРЕНЦИАЛЬНОЙ ГЕОМЕТРИИ ПЯТИМЕРНЫХ КОМПЛЕКСОВ ДВУМЕРНЫХ ПЛОСКОСТЕЙ ПРОЕКТИВНОГО ПРОСТРАНСТВА P5

The article focuses on the differential geometry  of 5-dimensional complexes  C5 of 2-dimensional planes in the projective  space  P5 that contains a finite  number of developable surfaces. This article relates to researches on the projective differential geometry based on E. Cartan’s moving frame method and the method of exterior differential forms. These methods make it possible to study the differential geometry of submanifolds of different dimensions of a Grassmann manifold from a single viewpoint and also extend the results to wider classes of manifolds of multidimensional planes. In order to study such submanifolds, we apply the Grassmann mapping of the manifold  G(2, 5) onto the 9-dimensional algebraic manifold Ω(2, 5) of the space P19. The main task of the differential geometry of submanifolds of Grassmann manifolds is to carry out uniform classifications of various classes of such submanifolds, determine their structure and define the degree of the arbitrariness of their existence and also to research properties of submanifolds of various classes. Intersection of the algebraic manifold Ω(2, 5) with its tangent space TlΩ(2, 5) represents the Segre cone Cl(3, 3). This 5-dimensional cone carries two sets of plane 3-dimensional generatrices intersecting in straight lines. The projectivization PBl(2) of this cone is the Segre manifold Sl(2, 2). The Segre manifold Sl(2, 2) is invariant under projective transformations of P8 = PTlΩ(2, 5), which is the projectivization with center at the point l of the tangent space TlΩ(2, 5) to the algebraic manifold Ω(2, 5). The Segre manifold Sl(2, 2) is used for classification of the submanifolds of Grassmann manifold G(2, 5) under consideration, as well as for interpretation of their properties  in the projective algebraic manifold terms.  The classification  of submanifolds of the Grassmann manifold G(2, 5) is  based on  various  configurations of the plane PTlΩ(2, 5) and on the Segre manifold Sl(2, 2).  The goal of this article  is to prove geometrically the theorem for determination of the order of the Segre invariant manifold Sl(2, 2).

C-projective geometry

We develop in detail the theory of c-projective geometry, a natural analogue of projective differential geometry adapted to complex manifolds. We realise it as a type of parabolic geometry and describe the associated Cartan or tractor connection. A Kaehler manifold gives rise to a c-projective structure and this is one of the primary motivations for its study. The existence of two or more Kaehler metrics underlying a given c-projective structure has many ramifications, which we explore in depth. As a consequence of this analysis, we prove the Yano-Obata conjecture for complete Kaehler manifolds: if such a manifold admits a one parameter group of c-projective transformations that are not affine, then it is complex projective space, equipped with a multiple of the Fubini-Study metric.

Higher symmetries of symplectic Dirac operator

We construct in projective differential geometry of the real dimension 2 higher symmetry algebra of the symplectic Dirac operator acting on symplectic spinors. The higher symmetry differential operators correspond to the solution space of a class of projectively invariant overdetermined operators of arbitrarily high order acting on symmetric tensors. The higher symmetry algebra structure corresponds to a completely prime primitive ideal having as its associated variety the minimal nilpotent orbit of $${\mathfrak {sl}}(3,{\mathbb {R}})$$ .

Projectively equivalent 2-dimensional superintegrable systems with projective symmetries

This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves. We give a definition of projective equivalence of such systems, which may be considered the projective analog of (conformal) Stäckel equivalence (coupling constant metamorphosis). Then, we discuss the transformation behavior for projectively equivalent superintegrable systems and find that the potential on a projectively equivalent geometry can be reconstructed from a characteristic vector field. Moreover, potentials of projectively equivalent Hamiltonians follow a linear superimposition rule. The techniques are applied to several examples. In particular, we use them to classify, up to Stäckel equivalence, the superintegrable systems on geometries with one, non-trivial projective symmetry.

C-projective compactification; (quasi-)Kähler metrics and CR boundaries

For complete complex connections on almost complex manifolds we introduce a natural definition of compactification. This is based on almost c-projective geometry, which is the almost complex analogue of projective differential geometry. The boundary at infinity is a (possibly non-integrable) CR structure. The theory applies to almost Hermitian manifolds which admit a complex metric connection of minimal torsion, which means that they are quasi-K\"ahler in the sense of Gray-Hervella; in particular it applies to K\"ahler and nearly K\"ahler manifolds. Via this canonical connection, we obtain a notion of c-projective compactification for quasi-K\"ahler metrics of any signature. We describe an asymptotic form for metrics that is necessary and sufficient for c-projective compactness. This metric form provides local examples and, in particular, shows that the usual complete K\"ahler metrics associated to smoothly bounded, strictly pseudoconvex domains in ${\Bbb C}^n$ are c-projectively compact. For a smooth manifold with boundary and a complete quasi-K\"ahler metric $g$ on the interior, we show that if its almost c-projective structure extends smoothly to the boundary then so does its scalar curvature. We prove that in this case (and under some natural assumptions on the extension of $J$ to the boundary) $g$ is almost c-projectively compact if and only if this scalar curvature is non-zero on an open dense set of the boundary. In that case it is, along the boundary, locally constant and hence nowhere zero there. Finally we describe the asymptotics of the curvature, showing, in particular, that the canonical connection satisfies an asymptotic Einstein condition. Key to much of the development is a certain real tractor calculus for almost c-projective geometry, and this is developed in the article.

Projective geometry of Sasaki–Einstein structures and their compactification

We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate application, this is used to describe the projective compactification of indefinite (suitably) complete non-compact Sasaki-Einstein structures and to prove that the boundary at infinity is a Fefferman conformal manifold that thus fibres over a nondegenerate CR manifold (of hypersurface type). We prove that this CR manifold coincides with the boundary at infinity for the c-projective compactification of the K\"ahler-Einstein manifold that arises, in the usual way, as a leaf space for the defining Killing field of the given Sasaki-Einstein manifold. A procedure for constructing examples is given. The discussion of symplectic holonomy reductions of projective structures leads us moreover to a new and simplifying approach to contact projective geometry. This is of independent interest and is treated in some detail.

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.

Tractor calculus, BGG complexes, and the cohomology of cocompact Kleinian groups

For a compact, oriented, hyperbolic n-manifold (M,g), realised as M=Γ\\Hn where Γ is a torsion-free cocompact subgroup of SO(n,1), we establish and study a relationship between differential geometric cohomology on M and algebraic invariants of the group Γ. In particular for F an irreducible SO(n,1)-module, we show that the group cohomology with coefficients H•(Γ,F) arises from the cohomology of an appropriate projective BGG complex on M. This yields the geometric interpretation that H•(Γ,F) parameterises solutions to certain distinguished natural PDEs of Riemannian geometry, modulo the range of suitable differential coboundary operators. Viewed in another direction, the construction shows one way that non-trivial cohomology can arise in a BGG complex, and sheds considerable light on its geometric meaning. We also use the tools developed to give a new proof that H1(Γ,S0kRn+1)≠0 whenever M contains a compact, orientable, totally geodesic hypersurface. All constructions use another result that we establish, namely that the canonical flat connection on a hyperbolic manifold coincides with the tractor connection of projective differential geometry.

Surface theory in discrete projective differential geometry. I. A canonical frame and an integrable discrete Demoulin system

We present the first steps of a procedure which discretizes surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adopt compact forms. Based on a scaling symmetry which injects a parameter into the linear Gauss-Weingarten equations, we set down an algebraic classification scheme of discrete projective minimal surfaces which turns out to admit a geometric counterpart formulated in terms of discrete notions of Lie quadrics and their envelopes. In the case of discrete Demoulin surfaces, we derive a Bäcklund transformation for the underlying discrete Demoulin system and show how the latter may be formulated as a two-component generalization of the integrable discrete Tzitzéica equation which has originally been derived in a different context. At the geometric level, this connection leads to the retrieval of the standard discretization of affine spheres in affine differential geometry.

Benenti Tensors: A useful tool in Projective Differential Geometry

Abstract Two metrics are said to be projectively equivalent if they share the same geodesics (viewed as unparametrized curves). The degree of mobility of a metric g is the dimension of the space of the metrics projectively equivalent to g. For any pair of metrics (g, ḡ) on the same manifold one can construct a (1, 1)- tensor L(g, ḡ) called the Benenti tensor. In this paper we discuss some geometrical properties of Benenti tensors when (g, ḡ) are projectively equivalent, particularly in the case of degree of mobility equal to 2.

Vortices, Painlevé integrability and projective geometry

The first half of the thesis concerns Abelian vortices and Yang-Mills (YM) theory. It is proved that the 5 types of vortices recently proposed by Manton are symmetry reductions of (A)SDYM equations with suitable gauge groups and symmetry groups acting as isometries in a 4-manifold. As a consequence, the twistor integrability results of such vortices can be derived. It is presented a natural definition of their kinetic energy and thus the metric of the moduli space was calculated by the Samols' localisation method. Then, a modified version of the Abelian-Higgs model is proposed in such a way that spontaneous symmetry breaking and the Bogomolny argument still hold. The Painleve test, when applied to its soliton equations, reveals a complete list of its integrable cases. The corresponding solutions are given in terms of third Painleve transcendents and can be interpreted as original vortices on surfaces with conical singularity. The last two chapters present the following results in projective differential geometry and Hamiltonians of hydrodynamic-type (HT) systems. It is shown that the projective structures defined by the Painleve equations are not metrisable unless either the corresponding equations admit first integrals quadratic in first derivatives or they define projectively flat structures. The corresponding first integrals can be derived from Killing vectors associated to the metrics that solve the metrisability problem. Secondly, it is given a complete set of necessary and sufficient conditions for an arbitrary affine connection in 2D to admit, locally, 0, 1, 2 or 3 Killing forms. These conditions are tensorial and simpler than the ones in previous literature. By defining suitable affine connections, it is shown that the problem of existence of Killing forms is equivalent to the conditions of the existence of Hamiltonian structures for HT systems of two components.

О строении комплексов m–мерных плоскостей проективного пространства $P^n$, содержащих конечное число торсов

Статья посвящена дифференциальной геометрии подмногообразий многообразий G(m, n), m-мерных плоскостей проективного пространства $P^n$, содержащих конечное число торсов. Для исследования таких подмногообразий используется грассманово отображение многообразия G(m, n) m-мерных плоскостей проективного пространства $P^n$ на (m+1)(n−m)-мерное алгебраическое многообразие $\Omega(m, n)$ пространства $P^N$, где $N=\left(\begin{array}{c}m+1\\n+1\\\end{array}\right)-1$. Это отображение в сочетании с методом внешних форм Э. Картана и методом подвижного репера позволило определить зависимость строения изучаемых многообразий и конфигурации (m − 1)-мерных характеристических плоскостей и (m+ 1)-мерных касательных плоскостей торсов, принадлежащих рассматриваемым многообразиям. We consider the projective differential geometry of m-dimensional plane submanifolds of manifolds G(m, n) in projective space $P^n$ containing a finite number of developable surfaces. To study such submanifolds we use the Grassmann mapping of manifolds G(m, n) of m-dimensional planes in projective space $P^n$ to $(m + 1)(n-m)$-dimensional algebraic manifold $\Omega(m, n)$ of space $P^N$, where $N=\left(\begin{array}{c}m+1\\n+1\\\end{array}\right)-1$. This mapping combined with the method of external Cartan’s forms and moving frame method made it possible to determine the dependence of considered manifolds structure and the configuration of the (m − 1)-dimensional characteristic planes and (m + 1)-dimensional tangential planes of developable surfaces that belong to considered manifolds.

Prolongation of symmetric Killing tensors and commuting symmetries of the Laplace operator

We determine the space of commuting symmetries of the Laplace operator on pseudo-Riemannian manifolds of constant curvature and derive its algebra structure. Our construction is based on Riemannian tractor calculus, allowing us to construct a prolongation of the differential system for symmetric Killing tensors. We also discuss some aspects of its relation to projective differential geometry.

One-Dimensional Projective Structures, Convex Curves and the Ovals of Benguria and Loss

Benguria and Loss have conjectured that, amongst all smooth closed curves in $${\mathbb{R}^2}$$ of length 2π, the lowest possible eigenvalue of the operator $${L=-\Delta+\kappa^2}$$ is 1. They observed that this value was achieved on a two-parameter family, $${\mathcal{O}}$$ , of geometrically distinct ovals containing the round circle and collapsing to a multiplicity-two line segment. We characterize the curves in $${\mathcal{O}}$$ as absolute minima of two related geometric functionals. We also discuss a connection with projective differential geometry and use it to explain the natural symmetries of all three problems.

Projective compactifications and Einstein metrics

Abstract For complete affine manifolds we introduce a definition of compactification based on the projective differential geometry (i.e. geodesic path data) of the given connection. The definition of projective compactness involves a real parameter α called the order of projective compactness. For volume preserving connections, this order is captured by a notion of volume asymptotics that we define. These ideas apply to complete pseudo-Riemannian spaces, via the Levi-Civita connection, and thus provide a notion of compactification alternative to conformal compactification. For many orders α, we provide an asymptotic form of a metric which is sufficient for projective compactness of the given order, thus also providing many local examples. Distinguished classes of projectively compactified geometries of orders one and two are associated with Ricci-flat connections and non-Ricci-flat Einstein metrics, respectively. Conversely, these geometric conditions are shown to force the indicated order of projective compactness. These special compactifications are shown to correspond to normal solutions of classes of natural linear PDE (so-called first BGG equations), or equivalently holonomy reductions of projective Cartan/tractor connections. This enables the application of tools already available to reveal considerable information about the geometry of the boundary at infinity. Finally, we show that metrics admitting such special compactifications always have an asymptotic form as mentioned above.

Projective minimality for centroaffine minimal surfaces

Centroaffine minimal surfaces are considered as an interesting class of surfaces from the viewpoint of not only variational problems in centroaffine differential geometry but also integrable systems. Typical examples of centroaffine minimal surfaces are proper affine spheres centered at the origin when we regard them as centroaffine surfaces. On the other hand, the study of projective minimal surfaces has a long history in projective differential geometry. Typical examples of projective minimal surfaces are proper affine spheres again, and so-called Demoulin surfaces or Godeaux-Rozet surfaces. In this paper, we shall regard centroaffine surfaces as projective surfaces and study projective minimality of centroaffine minimal surfaces. Using the fact that any centroaffine minimal surfaces have a one-parameter family of deformation known as associated surfaces, we shall give a classification of indefinite centroaffine minimal surfaces whose associated surfaces are all projective minimal, which includes centroaffine surfaces with vanishing Tchebychev operator and those found by the first author before. We shall also show that any indefinite centroaffine minimal surface whose associated surfaces are all Godeaux-Rozet surfaces is a proper affine sphere.

Chow form for projective differential variety

In this paper, a generic intersection theorem in projective differential algebraic geometry is given. Then, the Chow form for an irreducible projective differential variety is defined and basic properties of the differential Chow form in affine differential case are shown to be valid for its projective differential counterpart. Finally, we apply the differential Chow form to a result of linear dependence over projective varieties given by Kolchin.

On the branch curve of a general projection of a surface to a plane

In this paper we prove that the branch curve of a general projection of a surface to the plane is irreducible, with only nodes and cusps. This is a basic result in surface theory, extremely useful in various applications. However, its proof, in this general setting, was so far lacking. Our approach substantially uses a powerful tool from projective differential geometry, i.e., the concept of focal schemes.

Higher Order Connections

The purpose of this article is to present the theory of higher order connections on vector bundles from a viewpoint inspired by projective differential geometry.