Euclidean Space Form Research Articles

Periodic solutions of Hilbert’s fourth problem

It is shown that a possibly irreversible [Formula: see text] Finsler metric on the torus, or on any other compact Euclidean space form, whose geodesics are straight lines is the sum of a flat metric and a closed [Formula: see text]-form. This is used to prove that if [Formula: see text] is a compact Riemannian symmetric space of rank greater than one and [Formula: see text] is a reversible [Formula: see text] Finsler metric on [Formula: see text] whose unparametrized geodesics coincide with those of [Formula: see text], then [Formula: see text] is a Finsler symmetric space.

The generalized Lichnerowicz problem: Uniformly quasiregular mappings and space forms

A uqr mapping of annn-manifoldMMis a mapping which is rational with respect to a bounded measurable conformal structure onMM. Remarkably, the only closed manifolds on which locally (but not globally) injective uqr mappings act are Euclidean space forms. We further characterize space forms admitting uniformly quasiregular self mappings and we show that the space forms admitting branched uqr maps are precisely the spherical space forms. We further show that every non-injective uqr map of a Euclidean space form is a quasiconformal conjugate of a conformal map. This is not true if the non-injective hypothesis is removed.

On holomorphic immersions into kähler manifolds of constant holomorphic sectional curvature

We study holomorphic immersions f:X →M from a complex manifoldX into a Kahler manifold of constant holomorphic sectional curvatureM, i.e. a complex hyperbolic space form, a complex Euclidean space form, or the complex projective space equipped with the Fubini-Study metric. ForX compact we show that the tangent sequence splits holomorphically if and only iff is a totally geodesic immersion. ForX not necessarily compact we relate an intrinsic cohomological invariantp(X) onX, viz. the invariant defined by Gunning measuring the obstruction to the existence of holomorphic projective connections, to an extrinsic cohomological invariant(f) measuring the obstruction to the holomorphic splitting of the tangent sequence. The two invariantsp(X) and?(f) are related by a linear map on cohomology groups induced by the second fundamental form. In some cases, especially whenX is a complex surface andM is of complex dimension 4, under the assumption thatX admits a holomorphic projective connection we obtain a sufficient condition for the holomorphic splitting of the tangent sequence in terms of the second fundamental form.

Tight surfaces in three-dimensional compact Euclidean space forms

In this paper we define and discuss tight surfaces - smooth or polyhedral - in three-dimensional compact Euclidean space forms and prove existence and non-existence results. It will be shown that all orientable and most of the non-orientable surfaces can be tightly immersed in any of these space forms.

Light-front-quantized QCD in a covariant gauge

The light-front (LF) canonical quantization of quantum chromodynamics in covariant gauges is discussed. The Dirac procedure is used to eliminate the constraints in the gauge-fixed front form theory quantum action and to construct the LF Hamiltonian formulation. The physical degrees of freedom emerge naturally. The propagator of the dynamical ${\ensuremath{\psi}}_{+}$ part of the free fermionic propagator in the LF quantized field theory is shown to be causal and not to contain instantaneous terms. Since the relevant propagators in the covariant gauge formulation are causal, rotational invariance---including the Coulomb potential in the static limit---can be recovered, avoiding the difficulties encountered in light-cone gauge. Wick rotation may also be performed allowing the conversion of momentum space integrals into Euclidean space forms. Some explicit computations are done in quantum electrodynamics to illustrate the equivalence of front form theory with the conventional covariant formulation. LF quantization thus provides a consistent formulation of gauge theory, despite the fact that the hyperplanes ${x}^{\ifmmode\pm\else\textpm\fi{}}=0$ used to impose boundary conditions constitute characteristic surfaces of a hyperbolic partial differential equation.

On the Cohomology of Galois Groups Determined by Witt Rings

Let F denote a field of characteristic different from two. In this paper we describe the mod2 cohomology of a Galois group GF (called the W-group of F) which is known to essentially characterize the Witt ring WF of anisotropic quadratic modules over F. We show that H*(GF, F2) contains the mod2 Galois cohomology of F and that its structure will reflect important properties of the field. We construct a space XF endowed with an action of an elementary abelian group E such that the computation of the cohomology of GF reduces to calculating the equivariant cohomology H*E(XF, F2). For the case of a field which is not formally real this amounts to computing the cohomology of an explicit Euclidean space form, an object which is interesting in its own right. We provide a number of examples and a substantial combinatorial computation for the cohomology of the universal W-groups.

Reconstruction of 3-submanifolds of large codimension in euclidean spaces from their gauss image

We present necessary and sufficient conditions for a regular three-dimensional manifold in the Grassmannian manifoldG(m, m+3) to be the Gauss image of a regular 3-submanifold in (m+3)-dimensional Euclidean space form>4.

Local surgery obstructions and space forms

The space form problem, concerning the existence of free actions of finite groups on spheres, has now been substantially solved except in dimension three I-3, 4, 8, 11, 13, 14]. This paper gives an effective method for the uniqueness question: which homotopy types are realized by such actions. Our results are fairly complete for 2-hyperelementary periodic groups and give new necessary conditions for actions even on S 3. In particular Corollary C below eliminates certain homotopically non-linear actions which presented an obstacle to knowing precise dimensional bounds (see [4; Conjecture D]). The methods also apply to the euclidean space form problem of semi-free actions on I( "+k fixing I( k [2] where the linear model is a free representation direct sum with a trivial representation. A free action on S'- 1 yields a semi-free topological action on l(" by "coning". The method involves the calculation of a certain local surgery obstructions (w167 2, 3). The basic cases are the 2-hyperelementary type I groups: extensions of the form l ~ Z/m~ ~--, Z/2k--> l where m is odd. We write o- for a Sylow 2-subgroup of ~z (and identify it with Z/2k). Let t be the twisting defining the extension, and let t(cr) = Im(t: Z/2 k --* Aut(Z/m)) -~ 7/./2 z.

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 ϱ 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 π 1 onto ϱ 2, there is a diffeomorphism h: X 1→ X 2 such that h( ρ 1( r) x)= ρ 2( f( r) h( x). As an application, we try to decide the structure of affine motions of some euclidean space R n . First we verify the conjecture of [17, § 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 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( R n ) if they are isomorphic (cf.[8]).

Compact Euclidean space forms

Compact Euclidean space forms

The Calabi construction for compact Ricci flat Riemannian manifolds

1. The main result and some consequences. In 1956 E. Calabi [6] attacked the classification problem of compact euclidean space forms by means of a special construction, called the Calabi construction (see Wolf [14, p. 124]). Here we announce that the construction can be extended to compact riemannian manifolds whose Ricci curvature tensor is zero (Ricci flat). Of course, it is not known if there exist any Ricci flat nonflat compact riemannian manifolds, and in fact a search for such manifolds was the original motivation for our study. However, as a consequence of our extension of Calabi's result we reduce the question of existence of a compact nonflat Ricci flat manifold to the simply connected, connected case. In any case, we essentially reduce the construction of compact Ricci flat manifolds to the lower-dimensional case together with the case of first Betti number zero. As a further consequence of our construction we extend one of the Bieberbach theorems [4], [14, Theorem 3.3.1] from the flat to the Ricci flat case (Theorem 1.4) and give various sufficient topological conditions for a Ricci flat manifold to be flat. Our main result is the following: