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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]