arXiv subjects
Publications and source records attributed to Steven T. Tschantz.
In this paper, we analyze the two geometrical passages in Plato's Meno, (81c -- 85c) and (86e4 -- 87b2), from the points of view of a geometer in Plato's time and today. We give, in our opinion, a complete explanation of the difficult second geometrical passage. Our explanation solves an ingenious geometry puzzle that has baffled readers of Plato's Meno for over 2,400 years.
In this note we show that every integer is the signature of a non-compact, oriented, hyperbolic 4-manifold of finite volume, and give some partial results on the geography of such manifolds. The main ingredients are a theorem of Long and Reid, and the explicit construction of a hyperbolic 24-cell manifold with some special topological properties.
In this paper, we develop general techniques for computing the G-index of a closed, spin, hyperbolic 2- or 4-manifold, and apply these techniques to compute the G-index of the fully symmetric spin structure of the Davis hyperbolic 4-manifold.
A closed connected hyperbolic $n$-manifold bounds geometrically if it is isometric to the geodesic boundary of a compact hyperbolic $(n+1)$-manifold. A. Reid and D. Long have shown by arithmetic methods the existence of infinitely many manifolds that bound geometrically in every dimension. We construct here infinitely many explicit examples in dimension $n=3$ using right-angled dodecahedra and $120$-cells and a simple colouring technique introduced by M. Davis and T. Januszkiewicz. Namely, for every $k\geqslant 1$, we build an orientable compact closed $3$-manifold tessellated by $16k$ right-angled dodecahedra that bounds a $4$-manifold tessellated by $32k$ right-angled $120$-cells. A notable feature of this family is that the ratio between the volumes of the $4$-manifolds and their boundary components is constant and, in particular, bounded.
In this paper, we develop the theory for classifying all the geometric fibrations of compact, connected, flat $n$-orbifolds, over a 1-orbifold, up to affine equivalence. We apply our classification theory to classify all the geometric fibrations of compact, connected, flat $2$-orbifolds, over a 1-orbifold, up to affine equivalence. This paper is an essential part of our project to give a geometric proof of the classification of all closed flat 4-manifolds.
In this paper, we describe all the hyperbolic 24-cell 4-manifolds with exactly one cusp. There are four of these manifolds up to isometry. These manifolds are the first examples of one-cusped hyperbolic 4-manifolds of minimum volume.
This paper is a continuation of our previous paper, Co-Seifert fibrations of compact flat orbifolds, in which we developed the theory for classifying geometric fibrations of compact, connected, flat $n$-orbifolds, over a 1-orbifold, up to affine equivalence. In this paper, we apply our theory to classify all the geometric fibrations of compact, connected, flat $3$-orbifolds, over a 1-orbifold, up to affine equivalence.
In this paper, we study multiply transitive actions of the group of isometries of a cusped finite-volume hyperbolic 3-manifold on the set of its cusps. In particular, we prove a conjecture of Vogeler that there is a largest $k$ for which such $k$-transitive actions exist, and that for each $k \geq 3$, there is an upper bound on the possible number of cusps.
In this paper we prove that there is a direct relationship between Salem numbers and translation lengths of hyperbolic elements of arithmetic hyperbolic groups that are determined by a quadratic form over a totally real number field. As an application we determine a sharp lower bound for the length of a closed geodesic in a noncompact arithmetic hyperbolic n-orbifold for each dimension n. We also discuss a "short geodesic conjecture", and prove its equivalence with "Lehmer's conjecture" for Salem numbers.
Beside simplices, $n$-cubes form an important class of simple polyhedra. Unlike hyperbolic Coxeter simplices, hyperbolic Coxeter $n$-cubes are not classified. We show that there is no hyperbolic Coxeter $n$-cube for $n\geq~6$, and provide a full classification for $n\leq 5$. Our methods, which are essentially of combinatorial and algebraic nature, can be (and have been successfully) implemented in a symbolic computation software such as Mathematica$^\circledR$.
In this paper we use the G-spin theorem to show that the Davis hyperbolic 4-manifold admits harmonic spinors. This is the first example of a closed hyperbolic 4-manifold that admits harmonic spinors. We also explicitly describe the Spinor bundle of a spin hyperbolic 2- or 4-manifold and show how to calculated the subtle sign terms in the G-spin theorem for an isometry, with isolated fixed points, of a closed spin hyperbolic 2- or 4-manifold.
In this paper we prove that for each dimension $n$ there are only finitely many isomorphism classes of pairs of groups $(Γ,\mathrm{N})$ such that $Γ$ is an $n$-dimensional crystallographic group and $\mathrm{N}$ is a normal subgroup of $Γ$ such that $Γ/\mathrm{N}$ is a crystallographic group.
In this paper we describe the classification of all the geometric fibrations of a closed flat Riemannian 4-manifold over a 1-orbifold.
Let G be an n-dimensional crystallographic group (n-space group). If G is a Z-reducible, then the flat n-orbifold E^n/G has a nontrivial fibered orbifold structure. We prove that this structure can be described by a generalized Calabi construction, that is, E^n/G is represented as the quotient of the Cartesian product of two flat orbifolds under the diagonal action of a structure group of isometries. We determine the structure group and prove that it is finite if and only if the fibered orbifold structure has an orthogonally dual fibered orbifold structure. A geometric fibration of E^n/G corresponds to a space group extension 1 -> N -> G -> G/N -> 1. We give a criterion for the splitting of a space group extension in terms of the structure group action that is strong enough to detect the splitting of all the space group extensions corresponding to the standard Seifert fibrations of a compact, connected, flat 3-orbifold. If G is an arbitrary n-space group, we prove that the group Isom(E^n/G) of isometries of E^n/G is a compact Lie group whose component of the identity is a torus of dimension equal to the first Betti number of G. This implies that Isom(E^n/G) is finite if and only if G/[G,G] is finite. We describe how to classify all the geometric fibrations of compact, connected, flat n-orbifolds, over a 1-orbifold, up to affine equivalence. We apply our classification theory to the scientifically important case n = 3, and classify all the geometric fibrations of compact, connected, flat 3-orbifolds, over a 1-orbifold, up to affine equivalence.
In this paper, we compute the covolume of the group of units of the quadratic form f_d^n(x) = x_1^2 + x_2^2 + . . . + x_n^2 - d x_{n+1}^2 with d an odd, positive, square-free integer. Mcleod has determined the hyperbolic Coxeter fundamental domain of the reflection subgroup of the group of units of the quadratic form f_3^n. We apply our covolume formula to compute the volumes of these hyperbolic Coxeter polytopes.
By gluing together the sides of eight copies of an all-right angled hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of orientable hyperbolic 6-manifolds having the smallest possible volume.
In this paper, we prove that a normal subgroup N of an n-dimensional crystallographic group G determines a geometric fibered orbifold structure on the flat orbifold E^n/G, and conversely every geometric fibered orbifold structure on E^n/G is determined by a normal subgroup N of G, which is maximal in its commensurability class of normal subgroups of G. In particular, we prove that E^n/G is a fiber bundle, with totally geodesic fibers, over a b-dimensional torus, where b is the first Betti number of G. Let N be a normal subgroup of G which is maximal in its commensurability class. We study the relationship between the exact sequence 1 -> N -> G -> G/N -> 1 splitting and the corresponding fibration projection having an affine section. If N is torsion-free, we prove that the exact sequence splits if and only if the fibration projection has an affine section. If the generic fiber F = Span(N)/N has an ordinary point that is fixed by every isometry of F, we prove that the exact sequence always splits. Finally, we describe all the geometric fibrations of the orbit spaces of all 2- and 3-dimensional crystallographic groups building on the work of Conway and Thurston.
Many noncompact hyperbolic 3-manifolds are topologically complements of links in the 3-sphere. Generalizing to dimension 4, we construct a dozen examples of noncompact hyperbolic 4-manifolds, all of which are topologically complements of varying numbers of tori and Klein bottles in the 4-sphere. Finite covers of some of those manifolds are then shown to be complements of tori and Klein bottles in other simply-connected closed 4-manifolds. All the examples are based on a construction of Ratcliffe and Tschantz, who produced 1171 noncompact hyperbolic 4-manifolds of minimal volume. Our examples are finite covers of some of those manifolds.