Involutive moving frames II; The Lie-Tresse theorem

This paper continues the project, begun in [1], of harmonizing Cartan's classical equivalence method and the modern equivariant moving frame in a framework dubbed involutive moving frames. As an attestation of the fruitfulness of our framework, we obtain a new, constructive and intuitive proof of the Lie-Tresse theorem (Fundamental basis theorem) and a first general upper bound on the minimal number of generating differential invariants for Lie pseudo-groups. Further, we demonstrate the computational advantages of this framework by studying the equivalence problem for first order PDE in two independent variables and one dependent variable under point transformations.     

Affine differential geometry of surfaces in ℝ4

Employing the method of moving frames, i.e. Cartan's algorithm, we find a complete set of invariants for nondegenerate oriented surfacesM ² in ⁴ relative to the action of the general affine group on ⁴. The invariants found include a normal bundle, a quadratic form onM ² with values in the normal bundle, a symmetric connection onM ² and a connection on the normal bundle. Integrability conditions for these invariants are also determined. Geometric interpretations are given for the successive reductions to the bundle of affine frames overM ², obtained by using the method of moving frames, that lead to the aforementioned invariants. As applications of these results we study a class of surfaces known as harmonic surfaces, finding for them a complete set of invariants and their integrability conditions. Further applications involve the study of homogeneous surfaces; these are surfaces which are fixed by a group of affine transformations that act transitively on the surface. All homogeneous harmonic surfaces are determined.     

Invariants of Conformai and Projective Structures

While in Euclidean, equiaffine or centroaffine differential geometry there exists a unique, distinguished normalization of a regular hypersurface immersion x: M ⁿ → Aⁿ⁺¹, in the geometry of the general affine transformation group, there only exists a distinguished class of such normalizations, the class of relative normalizations. Thus, the appropriate invariants for speaking about affine hypersurfaces are invariants of the induced classes, e.g. the conformai class of induced metrics and the projective class of induced conormal connections. The aim of this paper is to study such invariants. As an application, we reformulate the fundamental theorem of affine differential geometry.     

Regelflächen im vierdimensionalen affinen Raum

In this paper we develop a theory of ruled surfaces of the four dimensional real affine space. An invariant moving frame is constructed using the halfinvariant methods of G. BOL whereby we get some geometric results. Furthermore a complete system of affine invariants is given and finally we consider ruled W-surfaces.

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Die vorliegende Arbeit enthält Teile der Dissertation des Verfassers (Universität Freiburg i. Br. 1977).     

On Flat Flag-Transitive c.c *-Geometries

We study flat flag-transitive c.c ^*-geometries. We prove that, apart from one exception related to Sym(6), all these geometries are gluings in the meaning of [6]. They are obtained by gluing two copies of an affine space over GF(2). There are several ways of gluing two copies of the n-dimensional affine space over GF(2). In one way, which deserves to be called the canonical one, we get a geometry with automorphism group G = 2²ⁿ · L _n(2) and covered by the truncated Coxeter complex of type D ₂ ⁿ . The non-canonical ways give us geometries with smaller automorphism group (G ≤ 2²ⁿ · (2 ^n?1)n) and which seldom (never ?) can be obtained as quotients of truncated Coxeter complexes.     

Symmetric complete sum-free sets in cyclic groups

We present constructions of symmetric complete sum-free sets in general finite cyclic groups. It is shown that the relative sizes of the sets are dense in [0, 1/3], answering a question of Cameron, and that the number of those contained in the cyclic group of order n is exponential in n. For primes p, we provide a full characterization of the symmetric complete sum-free subsets of ?_p of size at least (1/3?c)·p, where c > 0 is a universal constant.     

On locally trivial G a -actions

If the additive group of complex numbers acts algebraically on a normal affine variety, then the associated ring of invariants need not be finitely generated, but is an ideal transform of some normal affine algebra (Theorem 1). We investigate such normal affine algebras in the case of a locally trivial action on a factorial variety. If the variety is a complex affine space and the ring of invariants is isomorphic to a polynomial ring, then the action is conjugate to a translation (Theorem 3). Equivalently, ifC ⁿ , is the total space for a principalG _a -bundle over some open subset ofC ^n–1 then the bundle is trivial. On the other hand, there is a locally trivialG _a -action on a normal affine variety with nonfinitely generated ring of invariants (Theorem 2).Supported in part by NSA Grant No. MDA904-96-1-0069     

Algorithms for Symmetric Differential Systems

Over-determined systems of partial differential equations may be studied using differential—elimination algorithms, as a great deal of information about the solution set of the system may be obtained from the output. Unfortunately, many systems are effectively intractable by these methods due to the expression swell incurred in the intermediate stages of the calculations. This can happen when, for example, the input system depends on many variables and is invariant under a large rotation group, so that there is no natural choice of term ordering in the elimination and reduction processes. This paper describes how systems written in terms of the differential invariants of a Lie group action may be processed in a manner analogous to differential—elimination algorithms. The algorithm described terminates and yields, in a sense which we make precise, a complete set of representative invariant integrability conditions which may be calculated in a ``critical pair' completion procedure. Further, we discuss some of the profound differences between algebras of differential invariants and standard differential algebras. We use the new, regularized moving frame method of Fels and Olver [11], [12] to write a differential system in terms of the invariants of a symmetry group. The methods described have been implemented as a package in \MAPLE. The main example discussed is the analysis of the (2+1 )-d'Alembert—Hamilton system u_{xx}+u_{yy}- u_{zz}&=& f(u), u_x^2+u_y^2- u_z^2&=&1. (1) We demonstrate the classification of solutions due to Collins [7] for f\ne 0 using the new methods. October 13, 1999. Final version received: May 18, 2001.     

Elimination for Generic Sparse Polynomial Systems

We present a new probabilistic symbolic algorithm that, given a variety defined in an n-dimensional affine space by a generic sparse system with fixed supports, computes the Zariski closure of its projection to an ?-dimensional coordinate affine space with ?<n. The complexity of the algorithm depends polynomially on some combinatorial invariants associated to the supports.     

Curious Characterizations of Projective and Affine Geometries

We prove the following characterization theorem: If any three of the following four matroid invariants—the number of points, the number of lines, the coefficient of λ^n − 2 in the characteristic polynomial, and the number of three-element dependent sets—of a rank-n combinatorial geometry (or simple matroid) are the same as those of a rank-n projective geometry, then it is a projective geometry (of the same order). To do this, we use a lemma which is of independent interest: If H is a geometry in which all the lines have exactly ℓ − 1 or ℓ points and G is a geometry with at least three of the four matroid invariants the same as H, then all the lines in G also have exactly ℓ − 1 or ℓ points. An analogue of the characterization theorem holds for affine geometries. Our methods also yield inequalities amongst the four matroid invariants.     

On Ochoa's special matrices in matrix classes

It is well known that the ideal classes of an order Z[μ], generated over Z by the integral algebraic number μ, are in a bijective correspondence with certain matrix classes, that is, classes of unimodularly equivalent matrices with rational integer coefficients. If the degree of μ is ?3, we construct explicitly a particularly simple ideal matrix for an ideal which is a product of different prime ideals of degree 1. We obtain the following special n×n matrix (c_ij) in the matrix class corresponding to the ideal class of our ideal: c_i+1,_i=1(i=1,…,n?2); c_ij=0(?i?n, 1?j?n? 2, and i≠j+1); c_nj=0(j)=2,…,n?1). The remaining coefficients are given as explicit polynomials in an integer z which depends on the ideal. It is shown that the matrix class of every regular ideal class of Z[μ] contains a special matrix of this kind.     

The spectrum of Hill's equation

Letq be an infinitely differentiable function of period 1. Then the spectrum of Hill's operatorQ=?d ²/dx ²+q(x) in the class of functions of period 2 is a discrete series - ∞<λ₀<λ₁≦λ₂<λ₃≦λ₄<...<λ_2i?1≦λ_2i ↑∞. Let the numer of simple eigenvalues be 2n+1<=∞. Borg [1] proved thatn=0 if and only ifq is constant. Hochstadt [21] proved thatn=1 if and only ifq=c+2p with a constantc and a Weierstrassian elliptic functionp. Lax [29] notes thatn=m if¹ q=4k ² K ² m(m+1)sn ²(2Kx,k). The present paper studies the casen<∞, continuing investigations of Borg [1], Buslaev and Faddeev [2], Dikii [3, 4], Flaschka [10], Gardneret al. [12], Gelfand [13], Gelfand and Levitan [14], Hochstadt [21], and Lax [28–30] in various directions. The content may be summed up in the statement thatq is an abelian function; in fact, from the present standpoint, the whole subject appears as a part of the classical function theory of the hyperelliptic irrationality \(\ell (\lambda ) = \sqrt { - (\lambda - \lambda _0 )(\lambda - \lambda _1 )...(\lambda - \lambda _{2n} )} .\) The casen=∞ requires the development of the theory of abelian and theta functions for infinite genus; this will be reported upon in another place. Some of the results have been obtained independently by Novikov [34], Dubrovin and Novikov [6] and A. R. Its and V. B. Matveev [22].     

A construction of 2-designs from Steiner systems and extendable designs

We give a construction of a 2-(mn²+1,mn,(n+1)(mn−1)) design starting from a Steiner system S(2,m+1,mn²+1) and an affine plane of order n. This construction is applied to known classes of Steiner systems arising from affine and projective geometries, Denniston designs, and unitals. We also consider the extendability of these designs to 3-designs.     

Planar and affine spaces

In this note, we characterize finite three-dimensional affine spaces as the only linear spaces endowed with set Ω of proper subspaces having the properties (1) every line contains a constant number of points, say n, with n>2; (2) every triple of noncollinear points is contained in a unique member of Ω; (3) disjoint or coincide is an equivalence relation in Ω with the additional property that every equivalence class covers all points. We also take a look at the case n=2 (in which case we have a complete graph endowed with a set Ω of proper complete subgraphs) and classify these objects: besides the affine 3-space of order 2, two small additional examples turn up. Furthermore, we generalize our result in the case of dimension greater than three to obtain a characterization of all finite affine spaces of dimension at least 3 with lines of size at least 3.     

Amalgamation functors and boundary properties in simple theories

This paper continues the study of generalized amalgamation properties begun in [1], [2], [3], [5] and [6]. Part of the paper provides a finer analysis of the groupoids that arise from failure of 3-uniqueness in a stable theory. We show that such groupoids must be abelian and we link the binding group of the groupoids to a certain automorphism group of the monster model, showing that the group must be abelian as well. We also study connections between n-existence and n-uniqueness properties for various “dimensions” n in the wider context of simple theories. We introduce a family of weaker existence and uniqueness properties. Many of these properties did appear in the literature before; we give a category-theoretic formulation and study them systematically. Finally, we give examples of first-order simple unstable theories showing, in particular, that there is no straightforward generalization of the groupoid construction in an unstable context.     

On homotopy groups of quandle spaces and the quandle homotopy invariant of links

For a quandle X, the quandle space BX is defined, modifying the rack space of Fenn, Rourke and Sanderson (1995) [13], and the quandle homotopy invariant of links is defined in Z[π₂(BX)], modifying the rack homotopy invariant of Fenn, Rourke and Sanderson (1995) [13]. It is known that the cocycle invariants introduced in Carter et al. (2005) [3], Carter et al. (2003) [5], Carter et al. (2001) [6] can be derived from the quandle homotopy invariant.In this paper, we show that, for a finite quandle X, π₂(BX) is finitely generated, and that, for a connected finite quandle X, π₂(BX) is finite. It follows that the space spanned by cocycle invariants for a finite quandle is finitely generated. Further, we calculate π₂(BX) for some concrete quandles. From the calculation, all cocycle invariants for those quandles are concretely presented. Moreover, we show formulas of the quandle homotopy invariant for connected sum of knots and for the mirror image of links.     

Computation of Affine Covariants of Quadratic Bivariate Differential Systems

This paper deals with affine covariants of autonomous differential systems. The main result is the construction of a minimal system of generators of the algebra of affine covariants of quadratic bivariate differential systems which is helpful in qualitative and numerical study. To this end, we establish a theorem (true for general systems of dimension n and degree m) which provides a procedure of construction of systems of generators for affine covariants from those of center-affine invariants. After applying this theorem to the case n=m=2 we give the expansions of the obtained affine covariants in terms of center-affine covariants. All algorithms constitute the package SIB.     

Eigenvalues and invariants of tensors

A tensor is represented by a supermatrix under a co-ordinate system. In this paper, we define E-eigenvalues and E-eigenvectors for tensors and supermatrices. By the resultant theory, we define the E-characteristic polynomial of a tensor. An E-eigenvalue of a tensor is a root of the E-characteristic polynomial. In the regular case, a complex number is an E-eigenvalue if and only if it is a root of the E-characteristic polynomial. We convert the E-characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor, i.e., they are invariant under co-ordinate system changes. We call them principal invariants of that tensor. The maximum number of principal invariants of mth order n-dimensional tensors is a function of m and n. We denote it by d(m,n) and show that d(1,n)=1, d(2,n)=n, d(m,2)=m for m?3 and d(m,n)?mn−1+?+m for m,n?3. We also define the rank of a tensor. All real eigenvectors associated with nonzero E-eigenvalues are in a subspace with dimension equal to its rank.     

Properly discontinuous actions of subgroups in amenable algebraic groups and its application to affine motions

This note will concern properly discontinuous actions of subgroups in real algebraic groups on contractible manifolds. Let (π,X,ρ) be such an action, where ρ:π→ Diff(X) is a homomorphism. We assume that ? extends to a smooth action of a real algebraic group G containing π. If such π has a nontrivial radical (i.e., unique maximal normal solvable subgroup), then we can apply the method of Seifert construction [14],[17] to yield that the quotient π\X supports the structure of an injective Seifert fibering with typical (resp. exceptional) fiber diffeomorphic to a solv (resp. infrasolv)-manifold (when π acts freely). When G is an amenable algebraic group, we can say about a uniqueness property for such actions. Namely, let (π_i, X_i, ρ_i) be actions as above (i= 1,2). Then, given an isomorphism f of π₁ onto ?₂, there is a diffeomorphism h: X₁→X₂ such that h(ρ₁(r)x)=ρ₂(f(r)h(x).As an application, we try to decide the structure of affine motions of some euclidean space $\text{R}$ⁿ. First we verify the conjecture of [17, 4 5], i.e., a compact complete affinely flat manifold admits a maximal toral action if its fundamental group has a nontrivial center. Second, a compact complete affinity flat manifold whose fundamental group is virtually polycyclic supports the structure of an infrasolvmanifold. This structure varies depending on its solvable kernel (if it is abelian or nilpotent, it must be a euclidean space form or an infranilmanifold respectively). If a group of the affine group A(n) acts properly discontinuously and with compact quotient of $\text{R}$ⁿ, then it is called an affine crystallographic group. Finally, we can say so far as to a uniqueness property that two virtually polycyclic affine crystallographic groups are conjugate inside Diff($\text{R}$ⁿ) if they are isomorphic (cf.[8]).     

Extended affine Weyl groups and Frobenius manifolds

We define certain extensions of affine Weyl groups (distinct from these considered by K. Saito [S1] in the theory of extended affine root systems), prove an analogue of Chevalley Theorem for their invariants, and construct a Frobenius structure on their orbit spaces. This produces solutions F(t₁, ..., t_n) of WDVV equations of associativity polynomial in t₁, ..., t_n-1, exp t_n.