An Area Preserving Projection from the Regular Octahedron to the Sphere

In this paper, we propose an area preserving bijective map from the regular octahedron to the unit sphere ${\mathbb{S}^2}$ , both centered at the origin. The construction scheme consists of two steps. First, each face F _i of the octahedron is mapped to a curved planar triangle ${\mathcal{T}_i}$ of the same area. Afterwards, each ${\mathcal{T}_i}$ is mapped onto the sphere using the inverse Lambert azimuthal equal area projection with respect to a certain point of ${\mathbb{S}^2}$ . The proposed map is then used to construct uniform and refinable grids on a sphere, starting from any triangular uniform and refinable grid on the triangular faces of the octahedron.     

Locally homogeneous rigid geometric structures on surfaces

We study locally homogeneous rigid geometric structures on surfaces. We show that a locally homogeneous projective connection on a compact surface is flat. We also show that a locally homogeneous unimodular affine connection ${\nabla}$ on a two dimensional torus is complete and, up to a finite cover, homogeneous. Let ${\nabla}$ be a unimodular real analytic affine connection on a real analytic compact connected surface M. If ${\nabla}$ is locally homogeneous on a nontrivial open set in M, we prove that ${\nabla}$ is locally homogeneous on all of M.     

Lagrangian Curves in a 4-Dimensional Affine Symplectic Space

Lagrangian curves in \(\mathbb {R}^{4}\) entertain intriguing relationships with second order deformation of plane curves under the special affine group and null curves in a 3-dimensional Lorentzian space form. We provide a natural affine symplectic frame for Lagrangian curves. It allows us to classify Lagrangian curves with constant symplectic curvatures, to construct a class of Lagrangian tori in \(\mathbb {R}^{4}\) and determine Lagrangian geodesics.     

Planar nearrings on the Euclidean plane

Planar near-rings are generalized rings which can serve as coordinate domains for geometric structures in which each pair of nonparallel lines has a unique point of intersection. It is known that all planar nearrings can be constructed from regular groups of automorphisms of groups which can be viewed as the “action groups” of the planar nearring. In this article, we study planar nearrings whose additive group is \({(\mathbb{R}^n,+)}\) , in particular, n = 1 and 2. It is natural to study topological planar nearrings in this context, following ideas of the late Kenneth D. Magill, Jr. In the case of n = 1, we characterize all topological planar nearrings by their action groups \({(\mathbb{R}^*, \cdot)}\) or \({(\mathbb{R}^+, \cdot)}\) . For n = 2, these action groups and the circle group \({(\mathbb{U}, \cdot)}\) seem to be the most interesting cases, but the last case can be excluded completely. As a consequence, we obtain characterizations of the semi-homogeneous continuous mappings from \({\mathbb{R}^n}\) to \({\mathbb{R}}\) for n = 1 and 2. Such a mapping f enjoys the property that f(f(u)v) = f(u)f(v) for all \({u,v \in \mathbb{R}^n}\) . When \({f(\mathbb{R}^n) = \mathbb{R}^+}\) , f is a positive homogeneous mapping of degree 1.     

Affine Surfaces which are Both Affine Harmonic and Affine Maximal

We study affine surfaces which are both affine maximal and affine harmonic. We prove that an indefinite surface satisfying both conditions is affine equivalent to an open part $(u,{1\over 2}u^2,P_1(u)+\upsilon,P_2(u)+{1\over 2}\upsilon^2)$ , where P₁ and P₂ are arbitrary functions of one variable.     

A functional model associated with a generalized Nevanlinna pair

Let $ \mathcal{L} $ be a Hilbert space, and let $ \mathcal{H} $ be a Pontryagin space. For every self-adjoint linear relation $ \tilde{A} $ in $ \mathcal{H} \oplus \mathcal{L} $ , the pair $ \{ I + \lambda \psi (\lambda ),\,\psi (\lambda )\} $ where $ \psi (\lambda ) $ is the compressed resolvent of $ \tilde{A} $ , is a normalized generalized Nevanlinna pair. Conversely, every normalized generalized Nevanlinna pair is shown to be associated with some self-adjoint linear relation $ \tilde{A} $ in the above sense. A functional model for this selfadjoint linear relation $ \tilde{A} $ is constructed.     

Cyclic symmetry of the scaled simplex

Let $\mathcal{Z}_{m}^{k}$ consist of the m ^k alcoves contained in the m-fold dilation of the fundamental alcove of the type A _k affine hyperplane arrangement. As the fundamental alcove has a cyclic symmetry of order k+1, so does $\mathcal{Z}_{m}^{k}$ . By bijectively exchanging the natural poset structure of $\mathcal{Z}_{m}^{k}$ for a natural cyclic action on a set of words, we prove that $(\mathcal{Z}_{m}^{k},\prod_{i=1}^{k} \frac{1-q^{m i}}{1-q^{i}},C_{k+1})$ exhibits the cyclic sieving phenomenon.     

Generalized Killing spinors on spheres

We study generalized Killing spinors on round spheres \(\mathbb {S}^n\) . We show that on the standard sphere \(\mathbb {S}^8\) any generalized Killing spinor has to be an ordinary Killing spinor. Moreover, we classify generalized Killing spinors on \(\mathbb {S}^n\) whose associated symmetric endomorphism has at most two eigenvalues and recover in particular Agricola–Friedrich’s canonical spinor on 3-Sasakian manifolds of dimension 7. Finally, we show that it is not possible to deform Killing spinors on standard spheres into genuine generalized Killing spinors.     

CR-quadrics with a symmetry property

For non-degenerate CR-quadrics ${Q \subset \mathbb{C}^{n}}$ it is well known that the real Lie algebra ${\mathfrak{g} = \mathfrak{hol}(Q)}$ of all infinitesimal CR-automorphisms has a canonical grading ${\mathfrak{g} = \mathfrak{g}^{-2} \oplus\mathfrak{g}^{-1} \oplus\mathfrak{g}^{0} \oplus\mathfrak{g}^{1} \oplus\mathfrak{g}^{2}}$ . While the first three spaces in this grading, responsible for the affine automorphisms of Q, are always easy to describe this is not the case for the last two. In general, it is even difficult to determine the dimensions of ${\mathfrak{g}^{1}}$ and ${\mathfrak{g}^{2}}$ . Here we consider a class of quadrics with a certain symmetry property for which ${\mathfrak{g}^{1}, \mathfrak{g}^{2}}$ can be determined explicitly. The task then is to verify that there exist enough interesting examples. By generalizing the ?ilov boundaries of irreducible bounded symmetric domains of non-tube type we get a collection of basic examples. Further examples are obtained by ‘tensoring’ any quadric having the symmetry property with an arbitrary commutative (associative) unital *-algebra A (of finite dimension). For certain quadrics this also works if A is not necessarily commutative.     

Highly Incident Configurations with Chiral Symmetry

A geometric $k$ -configuration is a collection of points and straight lines in the plane so that $k$ points lie on each line and $k$ lines pass through this point. We introduce a new construction method for constructing $k$ -configurations with non-trivial dihedral or chiral (i.e., purely rotational) symmetry, for any $k \ge 3$ ; the configurations produced have $2^{k-2}$ symmetry classes of points and lines. The construction method produces the only known infinite class of symmetric geometric 7-configurations, the second known infinite class of symmetric geometric 6-configurations, and the only known 6-configurations with chiral symmetry.     

The subelliptic heat kernel on the CR sphere

We study the heat kernel of the sub-Laplacian $L$ on the CR sphere $\mathbb{S }^{2n+1}$ . An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we recover in a simple way the Green function of the conformal sub-Laplacian $-L+n^2$ that was obtained by Geller (J Differ Geom 15:417–435, 1980), and also get an explicit formula for the sub-Riemannian distance. The key point is to work in a set of coordinates that reflects the symmetries coming from the fibration $\mathbb{S }^{2n+1} \rightarrow \mathbb{CP }^n$ .     

The Cauchy problem for indefinite improper affine spheres and their Hessian equation

We give a conformal representation for indefinite improper affine spheres which solve the Cauchy problem for their Hessian equation. As consequences, we can characterize their geodesics and obtain a generalized symmetry principle. Then, we classify the helicoidal indefinite improper affine spheres and find a new family with geodesically complete non-flat affine metric. Moreover, we present interesting examples with singular curves and isolated singularities.     

Smooth Stable Planes

This paper deals with smooth stable planes which generalize the notion of differentiable (affine or projective) planes [7]. It is intended to be the first one of a series of papers on smooth incidence geometry based on the Habilitationsschrift of the author. It contains the basic definitions and results which are needed to build up a foundation for a systematic study of smooth planes. We define smooth stable planes, and we prove that point rows and line pencils are closed submanifolds of the point set and line set, respectively (Theorem (1.6)). Moreover, the flag space is a closed submanifold of the product manifold $P\times {\cal L}$ (Theorem (1.14)), and the smooth structure on the set P of points and on the set ${\cal L}$ of lines is uniquely determined by the smooth structure of one single line pencil. In the second section it is shown that for any point p \te P the tangent space T_pP carries the structure of a locally compact affine translation plane ${\cal A}_p$ , see Theorem (2.5). Dually, we prove in Section 3 that for any line $L \in {\cal L}$ the tangent space ${\rm T}_L{\cal L}$ together with the set ${\cal \rm S}_L=\lbrace {\rm T}_{L}{\cal L}_p\mid p \in L\rbrace$ gives rise to some shear plane. It turned out that the translation planes ${\cal A}_p$ are one of the most important tools in the investigation of smooth incidence geometries. The linearization theorems (3.9), (3.11), and (4.4) can be viewed as the main results of this paper. In the closing section we investigate some homogeneity properties of smooth projective planes.     

Rational preimages in families of dynamical systems

Let ${\phi}$ be a rational function of degree at least two defined over a number field k. Let ${a \in \mathbb{P}^1(k)}$ and let K be a number field containing k. We study the cardinality of the set of rational iterated preimages Preim ${(\phi, a, K) = \{x_{0} \in \mathbb{P}^1(K) | \phi^{N} (x_0) = a {\rm for some} N \geq 1\}}$ . We prove two new results (Theorems 2 and 4) bounding ${|{\rm Preim}(\phi, a, K)|}$ as ${\phi}$ varies in certain families of rational functions. Our proofs are based on unit equations and a method of Runge for effectively determining integral points on certain affine curves. We also formulate and state a uniform boundedness conjecture for Preim ${(\phi, a, K)}$ and prove that a version of this conjecture is implied by other well-known conjectures in arithmetic dynamics.     

Singular gradient flow of the distance function and homotopy equivalence

Let $M$ be a Riemannian manifold and let $\varOmega $ be a bounded open subset of $M$ . It is well known that significant information about the geometry of $\varOmega $ is encoded into the properties of the distance, $d_{\partial \varOmega }$ , from the boundary of $\varOmega $ . Here, we show that the generalized gradient flow associated with the distance preserves singularities, that is, if $x_0$ is a singular point of $d_{\partial \varOmega }$ then the generalized characteristic starting at $x_0$ stays singular for all times. As an application, we deduce that the singular set of $d_{\partial \varOmega }$ has the same homotopy type as $\varOmega $ .     

On the planar integrable differential systems

In this paper, we prove that the C ¹ planar differential systems that are integrable and non-Hamiltonian roughly speaking are C ¹ equivalent to the linear differential systems ${\dot u= u}$ , ${\dot v= v}$ . Additionally, we show that these systems have always a Lie symmetry. These results are improved for the class of polynomial differential systems defined in ${\mathbb{R}^2}$ or ${\mathbb{C}^2}$ .     

On the ∞-Norm of the Cubic form of Complete Hyperbolic Affine Hyperspheres

Let ${M_n \subset \mathbb R^{n+1}}$ be a complete hyperbolic affine hypersphere with mean curvature H, ${H < 0}$ , and let C be its cubic form. We derive a differential inequality and an upper bound on the scalar function ${||C||_{\infty}}$ defined by the fiber-wise maximum of the value of C on the unit sphere bundle of M. The bounds are attained for the affine hyperspheres which are asymptotic to a simplicial cone. The results have applications in conic optimization.     

Pairs of Projections: Geodesics,Fredholm and Compact Pairs

A pair \((P, Q)\) of orthogonal projections in a Hilbert space \( \mathcal{H} \) is called a Fredholm pair if $$\begin{aligned} QP : R(P) \rightarrow R(Q) \end{aligned}$$ is a Fredholm operator. Let \( \mathcal{F} \) be the set of all Fredholm pairs. A pair is called compact if \(P-Q\) is compact. Let \( \mathcal{C} \) be the set of all compact pairs. Clearly \( \mathcal{C} \subset \mathcal{F} \) properly. In this paper it is shown that both sets are differentiable manifolds, whose connected components are parametrized by the Fredholm index. In the process, pairs \(P, Q\) that can be joined by a geodesic (or equivalently, a minimal geodesic) of the Grassmannian of \( \mathcal{H} \) are characterized: this happens if and only if $$\begin{aligned} \dim (R(P)\cap N(Q))=\dim (R(Q)\cap N(P)). \end{aligned}$$