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We investigate the property of boundary rigidity for the projective structures associated to torsion-free affine connections on connected smooth manifolds with boundary. We show that these structures are generically boundary rigid, meaning that any automorphism of a generic projective structure that restricts to the identity on the boundary must itself be the identity. However, and in contrast with what happens for example for conformal structures, we show that there exist projective structures which are not boundary rigid. We characterise these non-rigid structures by the vanishing of a certain local projective invariant of the boundary. [ABSTRACT FROM AUTHOR]

We investigate the question: for which functions f (x 1 , ... , x n) , g (x 1 , ... , x n) the asymptotic expansion of the integral ∫ g (x 1 , ... , x n) e f (x 1 , ... , x n) + x 1 y 1 + ⋯ + x n y n ħ d x 1 ... d x n consists only of the first term. We reveal a hidden projective invariance of the problem which establishes its relation with geometry of projective hypersurfaces of the form { (1 : x 1 : ... : x n : f) } . We also construct various examples, in particular we prove that Kummer surface in P 3 gives a solution to our problem. [ABSTRACT FROM AUTHOR]

We describe all metrics geodesically compatible with a gl -regular Nijenhuis operator L. The set of such metrics is large enough so that a generic local curve γ is a geodesic for a suitable metric g from this set. Next, we show that a certain evolutionary PDE system of hydrodynamic type constructed from L preserves the property of γ to be a g-geodesic. This implies that every metric g geodesically compatible with L gives us a finite-dimensional reduction of this PDE system. We show that its restriction onto the set of g-geodesics is naturally equivalent to the Poisson action of R n on the cotangent bundle generated by the integrals coming from geodesic compatibility. [ABSTRACT FROM AUTHOR]

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) . Moreover, we show that, if a holonomy reduction to Sp (p , q) 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. [ABSTRACT FROM AUTHOR]

In this work, we examine the isoptic surfaces of line segments in the S 2 × R and H 2 × R geometries, which are from the 8 Thurston geometries. Based on the procedure first described in Csima and Szirmai (Results Math 78:194-19, 2023), we are able to give the isoptic surface of any segment implicitly. We rely heavily on the calculations published in Szirmai (Bul Acad Ştiinţe Repub Mold Mat 2:44–61, 2020; Q J Math 73:477–494, 2022). As a special case, we examine the Thales sphere in both geometries, which are called Thaloid. In our work we will use the projective model of S 2 × R and H 2 × R described by Molnár (Beitr Algebra Geom 38:261–288, 1997). [ABSTRACT FROM AUTHOR]

We construct all orthogonal separating coordinates in constant curvature spaces of arbitrary signature. Further, we construct explicit transformation between orthogonal separating and flat or generalised flat coordinates, as well as explicit formulas for the corresponding Killing tensors and Stäckel matrices. [ABSTRACT FROM AUTHOR]

We present a Fefferman-type construction from Lagrangian contact to split-signature conformal structures and examine several related topics. In particular, we describe the canonical curves and their correspondence. We show that chains and null-chains of an integrable Lagrangian contact structure are the projections of null-geodesics of the Fefferman space. Employing the Fermat principle, we realize chains as geodesics of Kropina (pseudo-Finsler) metrics. Using recent rigidity results, we show that 'sufficiently many' chains determine the Lagrangian contact structure. Separately, we comment on Lagrangian contact structures induced by projective structures and the special case of dimension three. [ABSTRACT FROM AUTHOR]

We generalise a construction of Landsberg, which associates certain Clifford algebra representations to Severi varieties. We thus obtain a new proof of Russo’s Divisibility Property for LQEL varieties whose secant varieties do not fill the ambient space. We also make an observation about the question of whether there exists a smooth, irreducible, non-degenerate, projective variety with secant deficiency δ > 8 . [ABSTRACT FROM AUTHOR]

In this paper we deal with Nil geometry, which is one of the homogeneous Thurston 3-geometries. We define the “surface of a geodesic triangle” using generalized Apollonius surfaces. Moreover, we show that the “lines” on the surface of a geodesic triangle can be defined by the famous Menelaus’ condition and prove that Ceva’s theorem for geodesic trianglesistruein Nil space. In our work we will use the projective model of Nil geometry described by E. Molnár in [6].

A multidimensional projective space with a marked point (center) is considered. On the manifold of projective frames of the given space that are adapted to the center, the action of the stabilizer of the center of the group of projective transformations is introduced. We prove that linear frames, i.e., bases of the tangent vector space of the projective space at its center, can be identified with orbits of the adapted projective frames with respect to the action of the kernel of the epimorphism of Lie groups, which assigns to each transformation from the stabilizer its differential at the center. Using a multidimensional generalization of the Desargues theorem, we obtain a criterion for two adapted projective frames to belong to the same orbit. [ABSTRACT FROM AUTHOR]

In the projective space Pn, we consider the Grassmann-like manifold Gr∗ (m, n) of centered planes and the associated principal fibration. We perform the normalization of the manifold Gr∗(m, n) by the normal fields of the first and second orders and the adaptation of the frame corresponding to these normals. Semi-normalized spaces of the first and second kind and the normalized space are examined. The dynamics of changes of the corresponding fibrations is established when passing from the space Gr∗ (m, n) to the normalized space. [ABSTRACT FROM AUTHOR]

After having investigated the geodesic and translation triangles and their angle sums in Sol and S L 2 R ~ geometries we consider the analogous problem in Nil space that is one of the eight 3-dimensional Thurston geometries. We analyze the interior angle sums of translation triangles in Nil geometry and we provide a new approach to prove that it can be larger than or equal to π . Moreover, for the first time in non-constant curvature Thurston geometries we have developed a procedure for determining the equations of Nil isoptic surfaces of translation-like segments and as a special case of this we examine the Nil translation-like Thales sphere, which we call Thaloid. In our work we will use the projective model of Nil described by Molnár (Beitr Algebra Geom 38(2):261–288, 1997). [ABSTRACT FROM AUTHOR]

We present a geometric construction and characterization of 2n-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson–Walker metric construction for n-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive a complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an explicit geometrically constructed Fefferman–Graham ambient metric and show the vanishing of the Q-curvature. [ABSTRACT FROM AUTHOR]

We consider multicomponent local Poisson structures of the form P 3 + P 1 , under the assumption that the third-order term P 3 is Darboux–Poisson and nondegenerate, and study the Poisson compatibility of two such structures. We give an algebraic interpretation of this problem in terms of Frobenius algebras and reduce it to classification of Frobenius pencils, i.e. linear families of Frobenius algebras. Then, we completely describe and classify Frobenius pencils under minor genericity conditions. In particular,we show that each Frobenuis pencil is a subpencil of a certain maximal pencil. These maximal pencils are uniquely determined by some combinatorial object, a directed rooted in-forest with edges and vertices labelled by numerical marks. They are also naturally related to certain pencils of Nijenhuis operators. We show that common Frobenius coordinate systems admit an elegant invariant description in terms of the corresponding Nijenhuis pencils. [ABSTRACT FROM AUTHOR]

On an almost complex manifold, a quasi-Kähler metric, with canonical connection in the c-projective class of a given minimal complex connection, is equivalent to a nondegenerate solution of the c-projectively invariant metrizability equation. For this overdetermined equation, replacing this maximal rank condition on solutions with a nondegeneracy condition on the prolonged system yields a strictly wider class of solutions with non-vanishing (generalized) scalar curvature. We study the geometries induced by this class of solutions. For each solution, the strict point-wise signature partitions the underlying manifold into strata, in a manner that generalizes the model, a certain Lie group orbit decomposition of C P m . We describe the smooth nature and geometric structure of each strata component, generalizing the geometries of the embedded orbits in the model. This includes a quasi-Kähler metric on the open strata components that becomes singular at the strata boundary. The closed strata inherit almost CR-structures and can be viewed as a c-projective infinity for the given quasi-Kähler metric. [ABSTRACT FROM AUTHOR]

Given a parabolic geometry, it is sometimes possible to find special metrics characterised by some invariant conditions. In conformal geometry, for example, one asks for an Einstein metric in the conformal class. Einstein metrics have the special property that their geodesics are distinguished, as unparameterised curves, in the sense of parabolic geometry. This property characterises the Einstein metrics. In this article, we initiate a study of corresponding phenomena for other parabolic geometries, in particular for the hypersurface CR and contact Legendrean cases. [ABSTRACT FROM AUTHOR]

In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets. It is shown that these nets are planar Koenigs nets. Moreover, we show that general Koenigs nets are characterised by the existence of a 1-parameter family of touching inscribed conics. It is shown that the lines of any grid of quadrilaterals with touching inscribed conics are tangent to a common conic. These grids can be constructed via polygonal chains that are inscribed in conics. The special case of billiards in conics corresponds to incircular nets. [ABSTRACT FROM AUTHOR]

It is an important question whether it is possible to put a geometry on a given manifold or not. It is well known that any simply connected closed manifold admitting a real projective structure must be a sphere. Therefore, any simply connected manifold M which is not a sphere (dim M ≥ 4) does not admit a real projective structure. Cooper and Goldman gave an example of a 3-dimensional manifold not admitting a real projective structure and this is the first known example. In this article, by generalizing their work, we construct a manifold M n with the infinite fundamental group Z 2 ∗ Z 2 , for any n ≥ 4 , admitting no real projective structure. [ABSTRACT FROM AUTHOR]

We prove entropy rigidity for finite volume strictly convex projective manifolds in dimensions ≥ 3 , generalizing the work of [1] to the finite volume setting. The rigidity theorem uses the techniques of Besson, Courtois, and Gallot's entropy rigidity theorem. It implies uniform lower bounds on the volume of any finite volume strictly convex projective manifold in dimensions ≥ 3 . [ABSTRACT FROM AUTHOR]

Weyl (Zur Infinitisimalgeometrie: Einordnung der projektiven und der konformen Auffasung, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Göttinger Akademie der Wissenschaften, Göttingen, 1921) demonstrated that for a connected manifold of dimension greater than 1, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl rigid and genericity of Weyl rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of sub-Riemannian metrics, i.e., when the only sub-Riemannian metric having the same sub-Riemannian geodesics, up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper (Jean et al. in Geom Dedic 203(1):279–319, 2019). [ABSTRACT FROM AUTHOR]

Demoulin surfaces in the real projective 3-space are investigated. Our result enables us to establish a generalized Weierstrass type representation for definite Demoulin surfaces by virtue of primitive maps into a certain semi-Riemannian 6-symmetric space. [ABSTRACT FROM AUTHOR]

We define Cayley structures as a field of Cayley's ruled cubic surfaces over a four dimensional manifold and motivate their study by showing their similarity to indefinite conformal structures and their link to differential equations and the theory of integrable systems. In particular, for Cayley structures an extension of certain notions defined for indefinite conformal structures in dimension four are introduced, e.g., half-flatness, existence of a null foliation, ultra-half-flatness, an associated pair of second order ODEs, and a dispersionless Lax pair. After solving the equivalence problem we obtain the fundamental invariants, find the local generality of several classes of Cayley structures and give examples. [ABSTRACT FROM AUTHOR]

In this paper, we consider discrete groups in PGLd(R) acting convex co‐compactly on a properly convex domain in real projective space. For such groups, we establish an analogue of the well‐known flat torus theorem for CAT(0) spaces. [ABSTRACT FROM AUTHOR]

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.

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 sl (3 , R) . [ABSTRACT FROM AUTHOR]

The lower-order cr-invariant variational problem for Legendrian curves in the 3-sphere is studied, and its Euler–Lagrange equations are deduced. Closed critical curves are investigated. Closed critical curves with non-constant cr-curvature are characterized. We prove that their cr-equivalence classes are in one-to-one correspondence with the rational points of a connected planar domain. A procedure to explicitly build all such curves is described. In addition, a geometrical interpretation of the rational parameters in terms of three phenomenological invariants is given. [ABSTRACT FROM AUTHOR]

We study invariant systems of PDEs defining Killing vector-valued forms, and then we specialize to Killing spinor-valued forms. We give a detailed treatment of their prolongation and integrability conditions by relating the pointwise values of solutions to the curvature of the underlying manifold. As an example, we completely solve the equations on model spaces of constant curvature producing brand-new solutions which do not come from the tensor product of Killing spinors and Killing–Yano forms. [ABSTRACT FROM AUTHOR]

In the spirit of Kobayashi's applications of methods of invariant metrics to questions of projective geometry, we introduce a projective analogue of the complex squeezing function. Using Frankel's work, we prove that for convex domains it stays uniformly bounded from below. In the case of strongly convex domains, we show that it tends to 1 at the boundary. This is applied to get a new proof of a projective analogue of the Wong–Rosay theorem. [ABSTRACT FROM AUTHOR]

We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on arbitrary parameter rectangles. For discrete planar quadrilateral nets, circular nets, Q ∗ -nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes. [ABSTRACT FROM AUTHOR]

In this paper, it is proved that a connected 3-dimensional Riemannian manifold or a closed connected semi-Riemannian manifold M n ( n > 1 ) admitting a projective vector field with a non-linearizable singularity is projectively flat. [ABSTRACT FROM AUTHOR]

A natural parametrization of smooth projective plane curves which tolerates the presence of sextactic points is the Forsyth–Laguerre parametrization. On a closed projective plane curve, which necessarily contains sextactic points, this parametrization is, however, in general not periodic. We show that by the introduction of an additional scalar parameter α ≤ 1 2 one can define a projectively invariant 2 π -periodic global parametrization on every simple closed convex sufficiently smooth projective plane curve without inflection points. For non-quadratic curves this parametrization, which we call balanced, is unique up to a shift of the parameter. The curve is an ellipse if and only if α = 1 2 , and the value of α is a global projective invariant of the curve. The parametrization is equivariant with respect to duality. [ABSTRACT FROM AUTHOR]

A (flat) affine 3-manifold is a 3-manifold with an atlas of charts to an affine space R 3 with transition maps in the affine transformation group Aff (R 3) . We will show that a connected closed affine 3-manifold is either an affine Hopf 3-manifold or decomposes canonically to concave affine submanifolds with incompressible boundary, toral π -submanifolds and 2-convex affine manifolds, each of which is an irreducible 3-manifold. It follows that if there is no toral π -submanifold, then M is prime. Finally, we prove that if a closed affine manifold is covered by a connected open set in R 3 , then M is irreducible or is an affine Hopf manifold. [ABSTRACT FROM AUTHOR]

We present a fundamental theory of curves in the affine plane and the affine space, equipped with the general-affine groups GA(2) = GL(2, ℝ) ⋉ ℝ2 and GA(3) = GL(3, ℝ) ⋉ ℝ3, respectively. We define general-affine length parameter and curvatures and show how such invariants determine the curve up to general-affine motions. We then study the extremal problem of the general-affine length functional and derive a variational formula. We give several examples of curves and also discuss some relations with equiaffine treatment and projective treatment of curves. [ABSTRACT FROM AUTHOR]

The generalized Feix–Kaledin construction shows that c-projective 2n-manifolds with curvature of type (1, 1) are precisely the submanifolds of quaternionic 4n-manifolds which are fixed-point set of a special type of quaternionic circle action. In this paper, we consider this construction in the presence of infinitesimal symmetries of the two geometries. First, we prove that the submaximally symmetric c-projective model with type (1, 1) curvature is a submanifold of a submaximally symmetric quaternionic model and show how this fits into the construction. We give conditions for when the c-projective symmetries extend from the fixed-point set of the circle action to quaternionic symmetries, and we study the quaternionic symmetries of the Calabi and Eguchi–Hanson hyperkähler structures, showing that in some cases all quaternionic symmetries are obtained in this way. [ABSTRACT FROM AUTHOR]

In this paper we study the interior angle sums of geodesic triangles in Nil geometry and prove that these can be larger, equal or less than π. We use for the computations the projective model of Nil introduced by Molnár (Beitr. Algebra Geom. 38(2):261-288, 1997). [ABSTRACT FROM AUTHOR]

We show that for any complete connected Kähler manifold, the index of the group of complex affine transformations in the group of c-projective transformations is at most two unless the Kähler manifold is isometric to complex projective space equipped with a positive constant multiple of the Fubini-Study metric. This establishes a stronger version of the recently proved Yano-Obata conjecture for complete Kähler manifolds. [ABSTRACT FROM AUTHOR]

We study the homogeneous artinian ideals of the polynomial ring K [ x , y , z ] generated by the homogeneous polynomials of degree d which are invariant under an action of the cyclic group Z / d Z , for any d ≥ 3 . We prove that they are all monomial Togliatti systems, and that they are minimal if the action is defined by a diagonal matrix having on the diagonal ( 1 , e , e a ) , where e is a primitive d -th root of the unity. We get a complete description when d is prime or a power of a prime. We also establish the relation of these systems with linear Ceva configurations. [ABSTRACT FROM AUTHOR]

In this note we extend the main result in [6] on artinian ideals failing Lefschetz properties, varieties satisfying Laplace equations and existence of suitable singular hypersurfaces. Moreover we characterize the minimal generation of ideals generated by powers of linear forms by the configuration of their dual points in the projective plane and we use this result to improve some propositions on line arrangements and Strong Lefschetz Property at range 2 in [6] . The starting point was an example in [3] . Finally we show the equivalence among failing SLP, Laplace equations and some unexpected curves introduced in [3] . [ABSTRACT FROM AUTHOR]

We consider in this paper the following questions: does the Jacobian ideal of a smooth hypersurface have the Weak Lefschetz Property? Does the Jacobian ideal of a smooth hypersurface have the Strong Lefschetz Property? We prove that if X is a hypersurface in P n of degree d > 2 , such that its singular locus has dimension at most n − 3 , then the ideal J ( X ) has the WLP in degree d − 2 . Moreover we show that if X is a hypersurface in P n of degree d > 2 , such that its singular locus has dimension at most n − 3 , then for every positive integer k < d − 1 the ideal J ( X ) has the SLP in degree d − k − 1 at range k . Finally we prove that if X ⊂ P n is a general hypersurface and J ( X ) its Jacobian ideal, then J ( X ) has the SLP. We present four famous line arrangements that give new examples of ideals failing SLP; they all arise from complex reflection groups. We conclude giving an infinite family of line arrangements that produce ideals failing SLP. [ABSTRACT FROM AUTHOR]

The paper is based on the lecture course “Metric projective geometry” which I conducted at the summer school “Finsler geometry with applications” at Karlovassi, Samos, in 2014, and at the workshop before the 8th seminar on Geometry and Topology of the Iranian Mathematical society at the Amirkabir University of Technology in 2015. The goal of this lecture course was to show how effective projectively invariant objects can be used to solve natural and named problems in differential geometry, and this paper also does it: I give easy new proofs to many known statements and also prove the following new statement: on a complete Riemannian manifold on nonconstant curvature, the index of the group of affine transformations in the group of projective transformations is at most two. [ABSTRACT FROM AUTHOR]

We consider the volume expansion of the Blaschke metric, which is a projectively invariant metric on a strictly convex domain in a locally flat projective manifold. When the boundary is even dimensional, we express the logarithmic coefficient L as the integral of affine invariants over the boundary. We also formulate an intrinsic geometry of the boundary as a conformal Codazzi structure and show that L gives a global conformal invariant of the boundary. [ABSTRACT FROM AUTHOR]

In this paper we study the set of projective maps between compact properly convex real projective manifolds. We show that this set contains only finitely many distinct homotopy classes and each homotopy class has the structure of a real projective manifold. When domain is irreducible and the target is strictly convex, our results imply that each non-trivial homotopy class contains at most one projective map. These results are motivated by the theory of holomorphic maps between compact complex manifolds. [ABSTRACT FROM AUTHOR]

A metric projective structure is a manifold equipped with the unparametrized geodesics of some pseudo-Riemannian metric. We make a comprehensive treatment of such structures in the case that there is a certain (well-motivated) algebraic restriction on the projective Weyl curvature, a nullity condition. The analysis is simplified by a fundamental and canonical 2-tensor invariant that we discover. It leads to a new canonical tractor connection for these geometries, which is defined on a rank [ABSTRACT FROM AUTHOR]

The classical Demoulin transformation is examined in the context of discrete differential geometry. We show that iterative application of the Demoulin transformation to a seed projective minimal surface generates a $${\mathbb {Z}}^2$$ lattice of projective minimal surfaces. Known and novel geometric properties of these Demoulin lattices are discussed and used to motivate the notion of lattice Lie quadrics and associated discrete envelopes and the definition of the class of discrete projective minimal and Q-surfaces (PMQ-surfaces). We demonstrate that the even and odd Demoulin sublattices encode a two-parameter family of pairs of discrete PMQ-surfaces with the property that one discrete PMQ-surface constitute an envelope of the lattice Lie quadrics associated with the other. [ABSTRACT FROM AUTHOR]