Conical Existence of Closed Curves on Convex Polyhedra

Let C be a simple, closed, directed curve on the surface of a convex polyhedron P. We identify several classes of curves C that "live on a cone," in the sense that C and a neighborhood to one side may be isometrically embedded on the surface of a cone Lambda, with the apex a of Lambda enclosed inside (the image of) C; we also prove that each point of C is "visible to" a. In particular, we obtain that these curves have non-self-intersecting developments in the plane. Moreover, the curves we identify that live on cones to both sides support a new type of "source unfolding" of the entire surface of P to one non-overlapping piece, as reported in a companion paper.Comment: 24 pages, 15 figures, 6 references. Version 2 includes a solution to one of the open problems posed in Version 1, concerning quasigeodesic loop

On organizing principles of Discrete Differential Geometry. Geometry of spheres

Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse

Gauss images of hyperbolic cusps with convex polyhedral boundary

We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2\pi$ is the metric of the Gauss image of some convex polyhedral cusp. This result is an analog of the Rivin-Hodgson theorem characterizing compact convex hyperbolic polyhedra in terms of their Gauss images. The proof uses a variational method. Namely, a cusp with a given Gauss image is identified with a critical point of a functional on the space of cusps with cone-type singularities along a family of half-lines. The functional is shown to be concave and to attain maximum at an interior point of its domain. As a byproduct, we prove rigidity statements with respect to the Gauss image for cusps with or without cone-type singularities. In a special case, our theorem is equivalent to existence of a circle pattern on the torus, with prescribed combinatorics and intersection angles. This is the genus one case of a theorem by Thurston. In fact, our theorem extends Thurston's theorem in the same way as Rivin-Hodgson's theorem extends Andreev's theorem on compact convex polyhedra with non-obtuse dihedral angles. The functional used in the proof is the sum of a volume term and curvature term. We show that, in the situation of Thurston's theorem, it is the potential for the combinatorial Ricci flow considered by Chow and Luo. Our theorem represents the last special case of a general statement about isometric immersions of compact surfaces.Comment: 55 pages, 17 figure

The closure constraint for the hyperbolic tetrahedron as a Bianchi identity

The closure constraint is a central piece of the mathematics of loop quantum gravity. It encodes the gauge invariance of the spin network states of quantum geometry and provides them with a geometrical interpretation: each decorated vertex of a spin network is dual to a quantized polyhedron in $\mathbb{R}^{3}$. For instance, a 4-valent vertex is interpreted as a tetrahedron determined by the four normal vectors of its faces. We develop a framework where the closure constraint is re-interpreted as a Bianchi identity, with the normals defined as holonomies around the polyhedron faces of a connection (constructed from the spinning geometry interpretation of twisted geometries). This allows us to define closure constraints for hyperbolic tetrahedra (living in the 3-hyperboloid of unit future-oriented spacelike vectors in $\mathbb{R}^{3,1}$) in terms of normals living all in $SU(2)$ or in $SB(2,\mathbb{C})$. The latter fits perfectly with the classical phase space developed for $q$-deformed loop quantum gravity supposed to account for a non-vanishing cosmological constant $\Lambda>0$. This is the first step towards interpreting $q$-deformed twisted geometries as actual discrete hyperbolic triangulations.Comment: 31 page

A Generalization of the Source Unfolding of Convex Polyhedra

We present a new method for unfolding a convex polyhedron into one piece without overlap, based on shortest paths to a convex curve on the polyhedron. Our “sun unfoldings” encompass source unfolding from a point, source unfolding from an open geodesic curve, and a variant of a recent method of Itoh, O’Rourke, and Vîlcu

Deformations and stability in complex hyperbolic geometry

This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sullivan. Despite Goldman-Millson-Yue rigidity results for such complex manifolds of infinite volume, we present different classes of such manifolds that allow non-trivial (quasi-Fuchsian) deformations and point out that such flexible manifolds have a common feature being Stein spaces. While deformations of complex surfaces from our first class are induced by quasiconformal homeomorphisms, non-rigid complex surfaces (homotopy equivalent to their complex analytic submanifolds) from another class are quasiconformally unstable, but nevertheless their deformations are induced by homeomorphisms

Combinatorics of embeddings

We offer the following explanation of the statement of the Kuratowski graph planarity criterion and of 6/7 of the statement of the Robertson-Seymour-Thomas intrinsic linking criterion. Let us call a cell complex 'dichotomial' if to every cell there corresponds a unique cell with the complementary set of vertices. Then every dichotomial cell complex is PL homeomorphic to a sphere; there exist precisely two 3-dimensional dichotomial cell complexes, and their 1-skeleta are K_5 and K_{3,3}; and precisely six 4-dimensional ones, and their 1-skeleta all but one graphs of the Petersen family. In higher dimensions n>2, we observe that in order to characterize those compact n-polyhedra that embed in S^{2n} in terms of finitely many "prohibited minors", it suffices to establish finiteness of the list of all (n-1)-connected n-dimensional finite cell complexes that do not embed in S^{2n} yet all their proper subcomplexes and proper cell-like combinatorial quotients embed there. Our main result is that this list contains the n-skeleta of (2n+1)-dimensional dichotomial cell complexes. The 2-skeleta of 5-dimensional dichotomial cell complexes include (apart from the three joins of the i-skeleta of (2i+2)-simplices) at least ten non-simplicial complexes.Comment: 49 pages, 1 figure. Minor improvements in v2 (subsection 4.C on transforms of dichotomial spheres reworked to include more details; subsection 2.D "Algorithmic issues" added, etc