Riemannian manifolds with structure groupG 2

Summary Riemannian manifolds with structure group G ₂ are 7-dimensional and have a distinguished 3-form. In this paper such manifolds are treated as analogues of almost Hermitian manifolds. Thus S ⁷ has structure group G ₂ just as S ⁶ is an almost Hermitian manifold. We study the covariant derivative of the fundamental 3-form as was done in [GH]for almost Hermitian manifolds.     

Twistor spaces of hypercomplex manifolds are balanced

A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkähler. In the paper “Non-Hermitian Yang–Mills connections” [13], Kaledin and Verbitsky proved that the twistor space of a hyperkähler manifold admits a balanced metric; these were first studied in the article “On the existence of special metrics in complex geometry” [17] by Michelsohn. In the present article, we review the proof of this result and then generalize it and show that twistor spaces of general compact hypercomplex manifolds are balanced.     

On the Geometry of Hermitian Matrices of Order Three Over Finite Fields

Some geometry of Hermitian matrices of order three over GF(q²) is studied. The variety coming from rank 2 matrices is a cubic hypersurface M₇³of PG(8,q ) whose singular points form a variety H corresponding to all rank 1 Hermitian matrices. BesideM₇³ turns out to be the secant variety of H. We also define the Hermitian embedding of the point-set of PG(2, q²) whose image is exactly the variety H. It is a cap and it is proved that PGL(3, q²) is a subgroup of all linear automorphisms of H. Further, the Hermitian lifting of a collineation of PG(2, q²) is defined. By looking at the point orbits of such lifting of a Singer cycle of PG(2, q²) new mixed partitions of PG(8,q ) into caps and linear subspaces are given.     

A characterization of the minimalbasis of the torus

D. König asks the interesting question in [7] whether there are facts corresponding to the theorem of Kuratowski which apply to closed orientable or non-orientable surfaces of any genus. Since then this problem has been solved only for the projective plane ([2], [3], [8]). In order to demonstrate that König’s question can be affirmed we shall first prove, that every minimal graph of the minimal basis of all graphs which cannot be embedded into the orientable surface f of genusp has orientable genusp+1 and non-orientable genusq with 1≦q≦2p+2. Then let f be the torus. We shall derive a characterization of all minimal graphs of the minimal basis with the nonorientable genusq=1 which are not embeddable into the torus. There will be two very important graphs signed withX ₈ andX ₇ later. Furthermore 19 graphsG ₁,G ₂, ...,G ₁₉ of the minimal basisM(torus, >₄) will be specified. We shall prove that five of them have non-orientable genusq=1, ten of them have non-orientable genusq=2 and four of them non-orientable genusq=3. Then we shall point out a method of determining graphs of the minimal basisM(torus, >₄) which are embeddable into the projective plane. Using the possibilities of embedding into the projective plane the results of [2] and [3] are necessary. This method will be called saturation method. Using the minimal basisM(projective plane, >₄) of [3] we shall at last develop a method of determining all graphs ofM(torus, >₄) which have non-orientable genusq≧2. Applying this method we shall succeed in characterizing all minimal graphs which are not embeddable into the torus. The importance of the saturation method will be shown by determining another graphG ₂₀≠G ₁,G ₂, ...,G ₁₉ ofM(torus, >₄).     

Killing Vector Fields and Twistor Forms on Generalized Sasakian Space Forms

We study generalized Sasakian space form M(f ₁, f ₂, f ₃) when (i) the Reeb vector field of the almost contact metric structure is Killing, (ii) the Ricci tensor satisfies Einstein-like conditions and (iii) the fundamental 2-form of the almost contact metric structure is a twistor form.     

Maximum nullity of outerplanar graphs and the path cover number

Let G=(V,E) be a graph with V={1,2,…,n}. Define S(G) as the set of all n×n real-valued symmetric matrices A=[a_ij] with a_ij≠0,i≠j if and only if ij∈E. By M(G) we denote the largest possible nullity of any matrix A∈S(G). The path cover number of a graph G, denoted P(G), is the minimum number of vertex disjoint paths occurring as induced subgraphs of G which cover all the vertices of G.There has been some success with relating the path cover number of a graph to its maximum nullity. Johnson and Duarte [5], have shown that for a tree T,M(T)=P(T). Barioli et al. [2], show that for a unicyclic graph G,M(G)=P(G) or M(G)=P(G)-1. Notice that both families of graphs are outerplanar. We show that for any outerplanar graph G,M(G)?P(G). Further we show that for any partial 2-path G,M(G)=P(G).     

Real Hypersurfaces in Complex Two-Plane Grassmannians

 The complex two-plane Grassmannian G ₂(C ^m+2 in equipped with both a K?hler and a quaternionic K?hler structure. By applying these two structures to the normal bundle of a real hypersurface M in G ₂(C ^m+2 one gets a one- and a three-dimensional distribution on M. We classify all real hypersurfaces M in G ₂ C ^m+2 , m≥3, for which these two distributions are invariant under the shape operator of M. Received 13 November 1996; in revised form 3 March 1997     

The Baer-invariant of a semidirect product

In 1972 K.I. Tahara [7,2, Theorem 2.2.5], using cohomological methods, showed that if a finite group is the semidirect product of a normal subgroup N and a subgroup T, then M(T) is a direct factor of M(G), where M(G) is the Schur-multiplicator of G and in the finite case, is the second cohomology group of G. In 1977 W. Haebich [1, Theorem 1.7] gave another proof using a different method for an arbitrary group G.In this paper we generalize the above theorem. We will show that scN_cM(T) is a direct factor of _cM(G), where _c[3, p. 102] is the variety of nilpotent groups of class at most c ≥ 1 and _cM(G) is the Baer-invariant of the group G with respect to the variety _c [3, p. 107].     

On the Constant Type of Almost Hermitian Manifolds

We suggest a most natural generalization of the notion of constant type for nearly Kählerian manifolds introduced by A. Gray to arbitrary almost Hermitian manifolds. We prove that the class of almost Hermitian manifolds of zero constant type coincides with the class of Hermitian manifolds. We show that the class of G ₁-manifolds of zero constant type coincides with the class of 6-dimensional G ₁-manifolds with a non-integrable structure. Finally, we prove that the class of normal G ₂-manifolds of nonzero constant type coincides with the class of 4-dimensional G ₂-manifolds with a nonintegrable structure.     

The projective rank of a Hermitian symmetric space: a geometric approach and consequences

The “Projective Rank” of a compact connected irreducible Hermitian symmetric space M has been defined as the maximal complex dimension of the compact totally geodesic complex submanifolds having positive holomorphic bisectional curvature with the induced K?hler metric. We present a geometric way to compute this invariant for the space M based on ideas developed in [1], [13] and [14]. As a consequence we obtain the following inequality relating the Projective Rank, Pr(M), the usual rank,rk(M), and the 2-number # (which is known to be equal to the Euler-Poincare characteristic in these spaces). Received: 6 June 2000 / Published online: 1 February 2002     

Compactifying coverings of 3-manifolds

If a finitely presented groupG is negatively curved, automatic or asynchronously automatic thenG has an asynchronously bounded “almost prefix closed” combing. Results in [Br₁] and [E] imply that the fundamental group of any closed 3-manifold satisfying Thurston's geometrization conjecture has an asynchronously bounded, almost prefix closed combing. MAIN THEOREM. IfM is a compactP ²-irreducible 3-manifold,π ₁ (M) has an asynchronously bounded, almost prefix closed combing, andH, a subgroup ofπ ₁ (M), is quasiconvex with respect to this combing, then the cover ofM corresponding toH is a missing boundary manifold.     

On almost hyperHermitian structures on Riemannian manifolds and tangent bundles

Some results concerning almost hyperHermitian structures are considered, using the notions of the canonical connection and the second fundamental tensor field h of a structure on a Riemannian manifold which were introduced by the second author. With the help of any metric connection on an almost Hermitian manifold M an almost hyperHermitian structure can be constructed in the defined way on the tangent bundle TM. A similar construction was considered in [6], [7]. This structure includes two basic anticommutative almost Hermitian structures for which the second fundamental tensor fields h ¹ and h ² are computed. It allows us to consider various classes of almost hyperHermitian structures on TM. In particular, there exists an infinite-dimensional set of almost hyperHermitian structures on TTM where M is any Riemannian manifold.     

Positive Twistor Bundle of a Kähler Surface

The aim of this paper is to characterize Kähler surfaces in terms oftheir positive twistor bundle. We prove that an oriented four-dimensionalRiemannian manifold (M, g) admits a complex structure J compatible with the orientation and such that (M, g, J is a Kähler manifold ifand only if the positive twistor bundle (Z ⁺(M), g _c ) admits a verticalKilling vector field.     

An operator representation for weighted spaces of vector valued holomorphic functions

For any weighted space HV(G) of holomorphic functions on an open set G ? ?^Nwith a topology stronger than that of uniform convergence on the compact sets and for any quasibarrelled space E we prove the topological isomorphism $HV(G,E_{b}^{\prime})={\cal L}_b(E,HV(G))$ and derive a similar, more complicated isomorphism for weighted spaces of continuous functions. This generalizes results of [3], [7] and [6] and should be compared with the ∈-product representations for the corresponding spaces of functions with o-growth conditions. At the end we also show the topological isomorphism HV₁(G₁, HV₂(G₂)) = H (V₁ ? V₂)(G₁ × G₂).     

Comparing the Zagreb indices for graphs with small difference between the maximum and minimum degrees

The first Zagreb index M₁(G) and the second Zagreb index M₂(G) of a (molecular) graph G are defined as M₁(G)=∑_u∈V(G)(d(u))² and M₂(G)=∑_uv∈E(G)d(u)d(v), where d(u) denotes the degree of a vertex u in G. The AutoGraphiX system [M. Aouchiche, J.M. Bonnefoy, A. Fidahoussen, G. Caporossi, P. Hansen, L. Hiesse, J. Lacheré, A. Monhait, Variable neighborhood search for extremal graphs. 14. The AutoGraphiX 2 system, in: L. Liberti, N. Maculan (Eds.), Global Optimization: From Theory to Implementation, Springer, 2005; G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs: 1 The AutoGraphiX system, Discrete Math. 212 (2000) 29-44; G. Caporossi, P. Hansen, Variable neighborhood search for extremal graphs. 5. Three ways to automate finding conjectures, Discrete Math. 276 (2004) 81-94] conjectured that M₁/n≤M₂/m (where n=|V(G)| and m=|E(G)|) for simple connected graphs. Hansen and Vuki?evi? [P. Hansen, D. Vuki?evi?, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165-168] proved that it is true for chemical graphs and it does not hold for all graphs. Vuki?evi? and Graovac [D. Vuki?evi?, A. Graovac, Comparing Zagreb M₁ and M₂ indices for acyclic molecules, MATCH Commun. Math. Comput. Chem. 57 (2007) 587-590] proved that it is also true for trees. In this paper, we show that M₁/n≤M₂/m holds for graphs with Δ(G)−δ(G)≤2 and characterize the extremal graphs, the proof of which implies the result in [P. Hansen, D. Vuki?evi?, Comparing the Zagreb indices, Croat. Chem. Acta 80 (2007) 165-168]. We also obtain the result that M₁/n≤M₂/m holds for graphs with Δ(G)−δ(G)≤3 and δ(G)≠2.     

Isoliertheit und Stabilität von Flächen konstanter mittlerer Krümmung

In the Sobolev space H^m(B,?³), B the open unit disc in ?², we consider the set M_n of all conformally parametrized surfaces of constant mean curvature H with exactly n simple interior branch points (and no others). We denote by M*_n the set of all xεM_n with the following properties:

1. in every branch point the geometrical condition K_G¦x_Z¦≡O holds (K_G is the Gauss curvature and x_z is the complex gradient of the surface x).
2. the corresponding boundary value problem Δh+×_z{²(2H²-K_G)h=O,^hδB=O, is uniquely solvable.

We prove then, that the manifold M*=UM*_n is open and dense in the set of all surfaces of constant mean curvature H and that all x εM*_n are isolated and stable solutions of the Plateau problem corresponding to their boundary curves. In addition, the submanifold M*_n contains exactly all surfaces x for which the space of Jacobi fields is transversal (with exception of the 3-dimensional space of conformai directions) to the tangent space T_xM_n.