Affinely equivalent complete flat manifolds

Let E _Aff(Γ,G, m) be the set of affine equivalence classes of m-dimensional complete flat manifolds with a fixed fundamental group Γ and a fixed holonomy group G. Let n be the dimension of a closed flat manifold whose fundamental group is isomorphic to Γ. We describe E _Aff(Γ,G, m) in terms of equivalence classes of pairs (ε, ρ), consisting of epimorphisms of Γ onto G and representations of G in ℝ ^m-n . As an application we give some estimates of card E _Aff(Γ,G, m).     

Stability of submanifolds with parallel mean curvature in calibrated manifolds

On a Riemannian manifold $ \bar M^{m + n} $ \bar M^{m + n} with an (m + 1)-calibration Ω, we prove that an m-submanifold M with constant mean curvature H and calibrated extended tangent space ℝH ⋇ TM is a critical point of the area functional for variations that preserve the enclosed Ω-volume. This recovers the case described by Barbosa, do Carmo and Eschenburg, when n = 1 and Ω is the volume element of $ \bar M $ \bar M . To the second variation we associate an Ω-Jacobi operator and define Ω-stability. Under natural conditions, we show that the Euclidean m-spheres are the unique Ω-stable submanifolds of ℝ ^m+n . We study the Ω-stability of geodesic m-spheres of a fibred space form M ^m+n with totally geodesic (m + 1)-dimensional fibres.     

A Generalization of Bochner’s Extension Theorem to Rough Tubes

This paper generalizes Bochner’s extension theorem to tubes X+iℝ ^m where the set X⊂ℝ ^m is not necessarily a manifold.     

On 2-almost contact tachibana manifolds

LeM be a (2m+2)-dimensional Riemannian manifold with two structure vector fieldsξ _r (r=2m+1, 2m+2) and letη ^r =ξ _r ^b be their corresponding covectors (or Pfaffians). These vector fields define onM a 2-almost contact structure. If the 2-formϕ=η ^2m+1∧η ^2m+2 is harmonic, then, following S. Tachibana [12],M is a Tachibana manifold and in this caseM is covered with 2 families of minimal submanifolds tangent toD ^⊥={ξ r} and its complementary orthogonal distributionD ^⊤. On such a manifold a canonical eigenfunction α of the Laplacian is associated. Since the corresponding eingenvalue is negative,M cannot be compact. Any horizontal vector fieldX orthogonal to α^# is a skew-symmetric Killing vector field (see [6]). Next, we assume that the Tachibana manifoldM under consideration is endowed with a framedf-structure defined by an endomorphism ϕ of the tangent bundleTM. Infinitesimal automorphisms of the symplectic form Ω _ϕ are obtained.     

Maxwell equations and the direction of the electromagnetic field

Let M₀ be the Minkowski space, let Λ₂(M₀) be the space of bivectors in M₀, and let G¹ ⊂ Λ₂(M₀) be the manifold of directions of the physical space, consisting of simple bivectors with square −1. A mapping F: U → Λ₂(M₀), U ⊂ ℝ⁴, satisfying the Maxwell equations is regarded as the tensor of an electromagnetic field in vacuum. The field is described on the basis of a special decomposition F = eω + h(*ω), where the mapping ω: U → G¹ is called the direction of the field, and e: U → (0, +∞) and h: U → ℝ are the electric and magnetic coefficients of the field. The Maxwell equations are reformulated in terms of ω, e, and h. Electromagnetic fields whose set of directions is a point or a one-dimensional subset of G¹ are considered. Bibliography: 7 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 329, 2005, pp. 118–146.     

Automorphismes des cônes convexes

One studies the subgroups of GL(m,ℝ) which preserve a properly convex cone of ℝ ^m and whose action on ℝ ^m is irreducible. In particular, one describes the Zariski closure of these subgroups. As an application, one describes the Zariski closure G of the subgroups of GL(m,ℝ) all of whose elements have nothing but positive eigenvalues. For instance, one can get the group G=GL(m,ℝ) if and only if m≠≡2 modulo4.

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Automorphismes des c?nes convexes

Résumé. On étudie les sous-groupes de GL(m,ℝ) qui préservent un c?ne convexe saillant de ℝ ^m et dont l’action sur ℝ ^m est irréductible. En particulier, on décrit les adhérences de Zariski possibles pour ces sous-groupes. Comme application, on décrit les adhérences de Zariski G possibles pour les sous-groupes de GL(m,ℝ) dont tous les éléments ont toutes leurs valeurs propres positives. Par exemple, le groupe G=GL(m,ℝ) convient si et seulement si m≠≡2 modulo4.

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Oblatum 22-I-1999 & 10-XI-1999?Published online: 21 February 2000     

Symmetric products of surfaces and the cycle index

We study some of the combinatorial structures related to the signature ofG-symmetric products of (open) surfacesSP _G ^m (M)=M ^m/G whereG ⊂S _m.The attention is focused on the question, what information about a surfaceM can be recovered from a symmetric productSP ⁿ(M). The problem is motivated in part by the study of locally Euclidean topological commutative (m+k,m)-groups, [16]. Emphasizing a combinatorial point of view we express the signature Sign(SP _G ^m (M))in terms of the cycle index ofG, a polynomial which originally appeared in Pólya enumeration theory of graphs, trees, chemical structures etc. The computations are used to show that there exist punctured Riemann surfacesM _g,k,M _g′,k′such that the manifoldsSP ^m(M _g,k)andSP ^m(M)_g′,k′)are often not homeomorphic, although they always have the same homotopy type provided 2 _g +k=2 _g′ +k′ andk,k′≥1. Supported by the Serbian Ministry for Science and Technology, Grant No. 1643.     

Existence of limits of maximal means

For the class II(ℝ ^m ) of continuous almost periodic functionsf: ℝ ^m → ℝ, we consider the problem of the existence of the limit

The van Kampen obstruction and its relatives

We review a cochain-free treatment of the classical van Kampen obstruction ϑ to embeddability of an n-polyhedron in ℝ²ⁿ and consider several analogs and generalizations of ϑ, including an extraordinary lift of ϑ, which has been studied by J.-P. Dax in the manifold case. The following results are obtained: (1) The mod 2 reduction of ϑ is incomplete, which answers a question of Sarkaria. (2) An odd-dimensional analog of ϑ is a complete obstruction to linkless embeddability (=“intrinsic unlinking”) of a given n-polyhedron in ℝ²ⁿ⁺¹. (3) A “blown-up” one-parameter version of ϑ is a universal type 1 invariant of singular knots, i.e., knots in ℝ³ with a finite number of rigid transverse double points. We use it to decide in simple homological terms when a given integer-valued type 1 invariant of singular knots admits an integral arrow diagram (= Polyak-Viro) formula. (4) Settling a problem of Yashchenko in the metastable range, we find that every PL manifold N nonembeddable in a given ℝ ^m , m ≥ $ \frac{{3(n + 1)}} {2} $ \frac{{3(n + 1)}} {2} , contains a subset X such that no map N → ℝ ^m sends X and N \ X to disjoint sets. (5) We elaborate on McCrory’s analysis of the Zeeman spectral sequence to geometrically characterize “k-co-connected and locally k-co-connected” polyhedra, which we embed in ℝ2 ^n−k for k < $ \frac{{n - 3}} {2} $ \frac{{n - 3}} {2} , thus extending the Penrose-Whitehead-Zeeman theorem.     

Hypersurfaces of prescribed mean curvature enclosing a given body

Given a smooth domain Ω in ℝ ^m+1 with compact closure and a smooth integrable functionh: ℝ ^m+1→ℝ satisfyingh(x)≥H _∂Ω (x) on ∂Ω whereH _∂ω denotes the mean curvature of ∂Ω calculated w.r.t. the interior unit normal we show that there is a setA⊂ℝ ^m+1 with the properties andH _∂A=h on ∂A.     

A Face of a Projective Triangulation Removed for Its Geometric Realizability

Let M be a map on a surface F ². A geometric realization of M is an embedding of F ² into a Euclidian 3-space ℝ³ with no self-intersection such that each face of M is a flat polygon. In Bonnington and Nakamoto (Discrete Comput. Geom. 40:141–157, 2008), it has been proved that every triangulation G on the projective plane has a face f such that the triangulation G−f on the M?bius band obtained from G by removing the interior of f has a geometric realization. In this paper, we shall characterize such a face f of G.     

Time-like Willmore surfaces in Lorentzian 3-space

Let R13 be the Lorentzian 3-space with inner product (, ). Let Q3 be the conformal compactification of R13, obtained by attaching a light-cone C∞ to R13 in infinity. Then Q3 has a standard conformal Lorentzian structure with the conformal transformation group O(3,2)/{±1}. In this paper, we study local conformal invariants of time-like surfaces in Q3 and dual theorem for Willmore surfaces in Q3. Let M (?) R13 be a time-like surface. Let n be the unit normal and H the mean curvature of the surface M. For any p ∈ M we define S12(p) = {X ∈ R13 (X - c(P),X - c(p)) = 1/H(p)2} with c(p) = P 1/H(p)n(P) ∈ R13. Then S12 (p) is a one-sheet-hyperboloid in R3, which has the same tangent plane and mean curvature as M at the point p. We show that the family {S12(p),p ∈ M} of hyperboloid in R13 defines in general two different enveloping surfaces, one is M itself, another is denoted by M (may be degenerate), and called the associated surface of M. We show that (i) if M is a time-like Willmore surface in Q3 with non-degenerate associated surface M, then M is also a time-like Willmore surface in Q3 satisfying M = M; (ii) if M is a single point, then M is conformally equivalent to a minimal surface in R13.     

Isometric immersions with homothetical Gauss map

Let f:M be an isometric immersion of an m-dimensional Riemannian manifold M into the n-dimensional Euclidean space. Its Gauss map g:MG _m ( ⁿ ) into the Grassmannian G _m ( ⁿ ) is defined by assigning to every point of M its tangent space, considered as a vector subspace of ⁿ . The third fundamental form b of f is the pull-back of the canonical Riemannian metric on G _m ( ⁿ ) via g. In this article we derive a complete classification of all those f (with flat normal bundle) for which the Gauss map g is homothetical; i.e. b is a constant multiple of the Riemannian metric on M. Using these results we furthermore classify all those f (with flat normal bundle) for which the third fundamental form b is parallel w.r.t. the Levi-Civita connection on M.     

A note on certain square functions on product spaces

We study certain square functions on product spaces Rn × Rm, whose integral kernels are obtained from kernels which are homogeneous in each factor Rn and Rm and locally in L(log L) away from Rn × {0} and {0} × Rm by means of polynomial distortions in the radial variable. As a model case, we obtain that the Marcinkiewicz integral operator is bounded on Lp(Rn × Rm)(P > 1) for Ω∈ e Llog L(Sn-1 × Sm-1) satisfying the cancellation condition.     

Discrete Calderón’s Identity,Atomic Decomposition and Boundedness Criterion of Operators on Multiparameter Hardy Spaces

In this paper we establish a discrete Calderón’s identity which converges in both L ^q (ℝ ^n+m ) (1<q<∞) and Hardy space H ^p (ℝ ⁿ ×ℝ ^m ) (0<p≤1). Based on this identity, we derive a new atomic decomposition into (p,q)-atoms (1<q<∞) on H ^p (ℝ ⁿ ×ℝ ^m ) for 0<p≤1. As an application, we prove that an operator T, which is bounded on L ^q (ℝ ^n+m ) for some 1<q<∞, is bounded from H ^p (ℝ ⁿ ×ℝ ^m ) to L ^p (ℝ ^n+m ) if and only if T is bounded uniformly on all (p,q)-product atoms in L ^p (ℝ ^n+m ). The similar result from H ^p (ℝ ⁿ ×ℝ ^m ) to H ^p (ℝ ⁿ ×ℝ ^m ) is also obtained.     

The Euler-Poincaré-Hopf theorem for flat connections in some transitive Lie algebroids

This paper is a continuation of [19], [21], [22]. We study flat connections with isolated singularities in some transitive Lie algebroids for which either ℝ or sl(2, ℝ) or so(3) are isotropy Lie algebras. Under the assumption that the dimension of the isotropy Lie algebra is equal to n + 1, where n is the dimension of the base manifold, we assign to any such isolated singularity a real number called an index. For ℝ-Lie algebroids, this index cannot be an integer. We prove the index theorem (the Euler-Poincaré-Hopf theorem for flat connections) saying that the index sum is independent of the choice of a connection. Multiplying this index sum by the orientation class of M, we get the Euler class of this Lie algebroid. Some integral formulae for indices are given.     

The structure of some classes of <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S ²ⁿ⁺¹ (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S ¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.     

Holomorphic Cliffordian functions

The aim of this paper is to put the foundations of a new theory of functions, called holomorphic Cliffordian, which should play an essential role in the generalization of holomorphic functions to higher dimensions. Let ℝ_0,2m+1 be the Clifford algebra of ℝ^2m+1 with a quadratic form of negative signature, be the usual operator for monogenic functions and Δ the ordinary Laplacian. The holomorphic Cliffordian functions are functionsf: ℝ^2m+2 → ℝ_0,2m+1, which are solutions ofDδ ^m f = 0. Here, we will study polynomial and singular solutions of this equation, we will obtain integral representation formulas and deduce the analogous of the Taylor and Laurent expansions for holomorphic Cliffordian functions. In a following paper, we will put the foundations of the Cliffordian elliptic function theory.     

On the Construction of <Emphasis Type="Italic">p</Emphasis>-Harmonic Morphisms and Conformal Actions

We produce p-harmonic morphisms by conformal foliations and Clifford systems. First, we give a useful criterion for a foliation on an m-dimensional Riemannian manifold locally generated by conformal fields to produce p-harmonic morphisms. By using this criterion we manufacture conformal foliations, with codimension not equal to p, which are locally the fibres of p-harmonic morphisms. Then we give a new approach for the construction of p-harmonic morphisms from R^m／{0} to R^n. By the well-known representation of Clifford algebras, we find an abundance of the new 2/3 （m ＋ 1）-harmonic morphism Ф： R^m／{0} → R^n where m = 2κδ（n - 1）.     

Rigidity of minimal submanifolds with flat normal bundle

Let M ⁿ (n ≥ 3) be an n-dimensional complete immersed $ \frac{{n - 2}} {n} $ \frac{{n - 2}} {n} -super-stable minimal submanifold in an (n + p)-dimensional Euclidean space ℝ ^n+p with flat normal bundle. We prove that if the second fundamental form of M satisfies some decay conditions, then M is an affine plane or a catenoid in some Euclidean subspace.