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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]

We compute the minimal and the maximal bound on the number of generators of a minimal smooth monomial Togliatti system of forms of degree d in $$n+1$$ variables, for any $$d\ge 2$$ and $$n\ge 2$$ . We classify the Togliatti systems with number of generators reaching the lower bound or close to the lower bound. We then prove that if $$n=2$$ (resp. $$n=2,3$$ ) all range between the lower and upper bound is covered, while if $$n\ge 3$$ (resp. $$n\ge 4$$ ) there are gaps if we only consider smooth minimal Togliatti systems (resp. if we avoid the smoothness hypothesis). We finally analyze for $$n=2$$ the Mumford-Takemoto stability of the syzygy bundle associated with smooth monomial Togliatti systems. [ABSTRACT FROM AUTHOR]

The mobility of a Kähler metric is the dimension of the space of metrics with which it is c-projectively equivalent. The mobility is at least two if and only if the Kähler metric admits a nontrivial hamiltonian 2-form. After summarizing this relationship, we present necessary conditions for a Kähler metric to have mobility at least three: its curvature must have nontrivial nullity at every point. Using the local classification of Kähler metrics with hamiltonian 2-forms, we describe explicitly the Kähler metrics with mobility at least three and hence show that the nullity condition on the curvature is also sufficient, up to some degenerate exceptions. In an appendix, we explain how the classification may be related, generically, to the holonomy of a complex cone metric. [ABSTRACT FROM AUTHOR]

Using tractor methods, we exhibit the local structure of the zero set of a twistor spinor in any metric signature. It is given as the image under the exponential map of a distinguished totally lightlike subspace. Based on these results we are able to construct starting from a conformal structure admitting a twistor spinor with zero a projective structure on its zero set in a natural way. The construction is presented both explicitly, using a fixed metric in the conformal class, and more abstractly on the level of associated parabolic Cartan geometries. Finally, we use known results about pseudo-Riemannian extensions of 2-dimensional projective structures to $$(2,2)$$ -ASD conformal structures to find first examples of non-conformally flat, non Riemannian manifolds admitting a twistor spinor with nontrivial zero set. [ABSTRACT FROM AUTHOR]

We present the theory of tensors with Young tableau symmetry as an efficient computational tool in dealing with the polynomial first integrals of a natural system in classical mechanics. We relate a special kind of such first integrals, already studied by Lundmark, to Beltrami's theorem about projectively flat Riemannian manifolds. We set the ground for a new and simple theory of the integrable systems having only quadratic first integrals. This theory begins with two centered quadrics related by central projection, each quadric being a model of a space of constant curvature. Finally, we present an extension of these models to the case of degenerate quadratic forms. [ABSTRACT FROM AUTHOR]

A vector field on a Kähler manifold is called c-projective if its flow preserves the J -planar curves. We give a complete local classification of Kähler real 4-dimensional manifolds that admit an essential c-projective vector field. An important technical step is a local description of 4-dimensional c-projectively equivalent metrics of arbitrary signature. As an application of our results we prove the natural analog of the classical Yano–Obata conjecture in the pseudo-Riemannian 4-dimensional case. [ABSTRACT FROM AUTHOR]