10th International Conference on Geometry and ... - Springer Link

J. Geom. 103 (2012), 347–366 c 2012 Springer Basel 0047-2468/12/020347-20 published online November 3, 2012 DOI 10.1007/s00022-012-0130-6

Journal of Geometry

10th International Conference on Geometry and Applications Varna (Bulgaria), September 3–9, 2011

The Conference took place in the International Home of Scientists “F. JoliotCurie”, St. Constantine and Helena, Bulgaria. The Scientific Committee which led the Conference consisted of W. Benz, R. Fritsch, H. Havlicek, S. Ivanov, H. Karzel, A. Kreuzer, H.-J. Kroll, Ch. Lozanov, M. Marchi, D. Mekerov, U. Simon, G. Stanilov (Chairman), N. Stephanidis, H. Wefelscheid, H. Zeitler. Laudations in honour of Momme Thomsen, Rudolf Fritsch and Chavdar Lozanov were given by H. Wefelscheid, C. Lozanov and G. Stanilov, respectively. The following are abstracts of talks presented in the Conference.

Murat Babaarslan, Yusuf Ali Tandogan (Bozok University, Turkey) Yusuf Yayli (Ankara University, Turkey) On Bertrand Curves and Constant Slope Surfaces According to Darboux Frame In [1] we studied constant slope surfaces and Bertrand curves in Euclidean 3space. We found parametrization of constant slope surfaces for spherical images a space curve. Furthermore, we investigated Bertrand curves corresponding to constant parameter curves of constant slope surfaces. In this work, we give some characterizations of Bertrand curves and constant slope surfaces with respect to Darboux frame. Subsequently, we express some interesting relations. Reference: [1] M. Babaarslan, Y. Yayli, The characterizations of constant slope surfaces and Bertrand curves, Int. J. Phys. Sci. 6, 1868–1875 (2011)

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Martin Belger (Universität Leipzig, Germany) Cavalieri Geometry – Estimation of Lattice Point Numbers of Non-Convex Domains A transformation C = id + f (·)c : Rn → Rn is said to be a Cavalieri transformation along the transformation direction c with the transformation function f : Rn → R if f (x + λc) = f (x), ∀λ ∈ R, x ∈ Rn . The total set L1 of Cavalierians C along all directions c in Rn does not form a group. But L1c ⊂ L1 along each fixed c and {L1 } do so. The invariant theory in resp. to the group L1c is called Cavalieri Geometry along c, in resp. to {L1 } General Cavalieri Geometry. Invariants under C are e.g. the volumes and the lattice point numbers (the latter perhaps asymptotic invariants) of the domains in Rn . These Geometries contain a copiousness of geometric-analytical facts and topics (Steiner symmetrizations as Cavalierians; Fréchet derivations of C, its linearization and matrix representation; eigenvalue characterization of C; affine C’s; symplectic and also Riemannian Geometry of Cavalierians, geodesics in resp. to the Cavalieri metric; C-convexifiable domains, asymptotic circle convexity; C-transforms of some geometric-analytical terms; isoperimetric property under C and many new other things). About the estimation of lattice point numbers of large non-convex domains we still don’t know much. But by the Cavalieri Geometries we discover a might class of such so-called C-convexifiable domains which allow the same estimation of their lattice point numbers by their areas or volumes as that of Gauß, Weyl and others. The investigations concern above all Cavalierians on the R2 . Of course, there are advantageous generalizations of Cavalierians for n ≥ 3 and the corresponding geometries are transferable also onto fiber bundles even within the framework of the measure theory.

Walter Benz (University of Hamburg, Germany) Hyperbolic Translations of Real Inner Product Spaces Let X be a real inner product space of (finite or infinite) dimension ≥ 2 and O(X) be its group of all surjective (hence bijective) orthogonal transformations of X. The hyperbolic distance of x, y ∈ X is defined to be the real number hyp(x, y) ≥ 0 satisfying cosh hyp(x, y) = 1 + x2 1 + y 2 − xy. The surjective solutions f : X → X of the functional equation hyp f (x), f (y) = hyp(x, y) for all x, y ∈ X

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are called hyperbolic motions of X. The identity mapping of X is defined to be a hyperbolic translation of X besides every hyperbolic motion μ with an existing a = 0 satisfying 0 = μ(x) − x ∈ Ra for all x ∈ X. The set T (X) of all hyperbolic translations of X is not a subgroup of the group M (X, hyp) of all hyperbolic motions of X. The following theorem can be proved. Theorem. Every μ ∈ M (X, hyp) has a representation μ = T · ω with uniquely determined T ∈ T (X) and uniquely determined ω ∈ O(X).

Albert Borbely (Kuwait University, Kuwait) The Omori-Yau Maximum Principle A Riemannian manifold M is said to satisfy the Omori-Yau maximum principle if for any C 2 bounded function g : M → R there is a sequence xn ∈ M , such that limn→∞ g(xn ) = supM g, limn→∞ |∇g(xn )| = 0 and lim supn→∞ Δg(xn ) ≤ 0. It is shown that if the Ricci curvature does not approach −∞ too fast the manifold satisfies the Omori-Yau maximum principle. This improves earlier necessary conditions. The given condition is quite optimal.

Natasha Danailova (University of Architecture, Civil Engineering and Geodesy, Bulgaria) Multiview Drawing We live in 3D-world. A photograph (drawing, picture) of the object shows well enough how it looks. The parallel or the central projections of the object on the picture plane are very close to the human sensitivities. This image helps the user to imagine how the object looks. But this image is not adequate for transmitting the details of the idea from the designer (or architect) to the producer (or the civil engineer). Before the object can be made, an exact description must be conveyed to the producer. This can be made by so called orthographic projection or multiview projection by the tools of the descriptive geometry. Another application is to reduce the real 3D object in a form, suitable for computer representation. It means to find the coordinates of the characteristic points of the object. It can be done by introduction of Cartesian coordinate system related with the object, and to find its orthographic projections-the orthogonal projections of the object on the coordinate planes.

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The following problems can be considered from this point of view: First problem: Given a pictorial drawing, draw the top, front and right side views in their conventional relationship. Second problem: Given a pictorial view, find the coordinates of the characteristic points of the object. Third problem: Given the top and the front views, draw the isometric drawing or the central projection. Fourth problem: Changing the position of the observer, find different views of the object. The descriptive geometry is among the most relevant subjects in technical and engineering education. The fundamentals of descriptive geometry are based on the projective geometry. Descriptive geometry shows some spatial relationships or mappings of one part of a mechanism to another part; or of one part of a surface to another part and so on. Using the tools of the descriptive geometry, the study of these relationships is a process of drawing the objects in a series of auxiliary views until the required image is found. The graphical solutions must be accurate and the drawings must be of high quality. It can be achieved on the worksheet by hand or using CAD system.

Iva Dokuzova (Plovdiv University, Bulgaria) Almost Conformal Transformation in a Four-dimensional Riemannian Manifold with an Additional Structure The main purpose of the present paper is to continue the investigations in [1] and [2]. In our case we consider a four dimensional Riemannian manifold M which admits a circulant metric g and a circulant afinor q. So, the first orders of the local coordinates gij , qij are as follows g1j =(A, B, C, B); q1j = (0, 1, 0, 0), A, B, C ∈ F M . We know from [D. Razpopov, On a class of special Riemannian manifolds, arXiv:math.DG/1106.2758] that q 4 = id, g(qx, qy) = g(x, y), x, y ∈ χM. We see that the tensor field f = gq 2 is a metric, and we define an almost conformal transformation by the relation g˜ = αg + βf , α, β ∈ F M . ˜ be the connections of g and g˜, respectively, and let Theorem. Let ∇ and ∇ ˜ ∇q = 0. Then ∇q = 0 if and only if grad α = grad βq 2 , grad β = −grad βq 2 .

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Let 0 < B < C < A, 0 < β < α; w ∈ Tp M . We construct: 1)

an infinite series of the almost conformally connected metrics g0 , g1 , . . . , gn , . . .

2) 3)

We prove that all metrics are positively defined; the corresponding series of the angles φ0 , φ1 , . . . , φn , . . .; where φn is the angle between w and qw with respect to gn ; the corresponding series of the angles ϕ0 , ϕ1 , . . . , ϕn , . . ., where ϕn is the angle between w and q 2 w with respect to gn .

We prove that limϕn = limφn = 0. This work is partially supported by project RS11-FMI-004 of the Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria. References: [1] G. Dzhelepov, D. Razpopov, I. Dokuzova, Almost conformal transformation in a class of Riemannian manifolds, Proc. of the Anniversary International Conference of FMI, Plovdiv, 2010, 125–128, arXiv:math.DG/1010.4975. [2] D. Razpopov, On a class of special Riemannian manifolds. arXiv:math.DG/1106.2758.

Radostina Encheva (University of Shumen, Bulgaria) Geometric Invariants with Respect to One Subgroup of the M¨ obius Group in the Plane The two-dimensional unit sphere S 2 ⊂ R3 , centered at the origin, is a model of the Gauss plane together with a point of infinity ∞. This model is realized via a stereographic projection π from the north pole onto the plane R2 through the equator. Making ∞ correspond to the north pole, π is bijective conformal map. Under this map the sphere S 2 is known as Riemann sphere. We have the following identifications S 2 ∼ = R2 ∪ {∞} ∼ = 1 1 C ∪ {∞} ∼ = J (C), where J (C) is one-dimensional projective space with respect to the field of complex numbers C. The group of rigid motions on the sphere S 2 coincides with the group of rotations SO(3) in R3 which preserve the sphere S 2 . This group induces on the plane via the stereoobius group. We describe F0 graphic projection π a subgroup F0 of the M¨ by matrixes and by quaternions, using the Hamilton’s theorem of representation of the group SO(3). We obtain an explicit form of the isomorphism F0 SO(3) of the three-parameter groups F0 and SO(3). Euclidean theory of the Frenet plane curves is transferred naturally on the Riemann sphere. We find invariants of the Frenet plane curves under the group F0 . A theorem that defines any Frenet plane curve up to transformation of the group F0 has been proved. We describe some classes of Frenet plane curves under F0 and apply computer system Mathematica to visualization.

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Liudmila Filipova (Technical University-Sofia, Branch Plovdiv, Bulgaria) Circulate Quadrate of an 8-Parametric Family of Surfaces The point (x11 , y11 , z11 ), where x11 = x20 + 2y0 z0 , y11 = y02 + 2x0 z0 , z11 = z02 + 2x0 y0 , is called circulate quadrate of the point (x0 , y0 , z0 ). We apply this definition for the family of surfaces defined parametrically by the equations: pu pu cos(qu), y0 = b sin(qu) + dv cos sin(qu) , x0 = a cos(qu) + cv cos 2 2 pu , 0 ≤ u ≤ 2π, −1 ≤ v ≤ 1 (1) z0 = k + ev sin 2 We remark that the classical M¨ obius strip belongs to this family (namely if a = 1, b = 1, c = 1, d = 1, k = 0, e = 1, p = 1, q = 1). For a given surface from this family we calculate the unit normal vector field n. Then we calculate along the closed curve v = 0, defined by vector parametrization x = (x0 , y0 , z0 )(v = 0). We found for any integer q and any odd integer p: x(v = 0, u = 0) = x(v = 0, u = 2π) = a2 e1 + k 2 e3 and n(v = 0, u = 0) = √1 (e2 − e3 ), n(v = 0, u = 2π) = −n(v = 0, u = 0) for non zero numbers 2 a, b, c, q. Thus we have proved the following Theorem. For any non zero numbers a, b, c, non zero integer q and odd integer p the circulate quadrate of the surface (1) is non-orientable. The same holds also for the M¨ obius strip.

Rudolf Fritsch (Ludwig-Maximilians-Universität M¨ unchen, Germany) Cabri versus Cinderella Demonstrated for the Loci of Some Triangle Centers Dynamical Geometry Software (DGS) is a modern tool for visualizing sophisticated geometric facts. There are several systems on the market which differ not only in the outer appearance but also with respect to the mathematical background. Together with my student Carolin Stromeder we compared Cabri, developed by Jean-Marie Laborde (Grenoble) and Cinderella due to J¨ urgen Richter-Gebert (M¨ unchen) and Ulrich Kortenkamp (Karlsruhe). There are 3612 triangle centers listed in Clark Kimberling’s Encyclopedia of Triangle Centers (http://fac\discretionary-ulty.evans\discretionary-ville.edu/ ck6/ency\discretionary-clo\discretionary-pe\discretionary-dia/ETC.html).

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Following an idea of Milan Koman(Praha) we looked for the loci of some of these centers for the one-parameter families of right-angled triangles with fixed hypotenuse. Among the triangle centers under our consideration are the Gergonne point X(7), the Schiffler point X(21) and the Napoleon points X(17), X(18). The results are algebraic curves which we compute by means of the Computer Algebra System (CAS) Maple.

Georgi Georgiev (University of Shumen, Bulgaria) Rational Curves and Surfaces in the Three-Sphere Rational space curves and rational surfaces are widely used in geometric modeling and computer graphics. Their parametric representations in Bernstein-Bézier form are convenient for many applications. Another technique for investigation of rational curves and surfaces is the quaternion algebra. Some special classes of such curves and surfaces can be fully described in term of quaternions. There is a one-to-one correspondence between the points of the four-dimensional Euclidean space and the elements of quaternion algebra. This correspondence gives a very simple presentation of the stereographic projection of the unit 3-sphere onto its equatorial hyperplane. The aim of the talk is to give explicit parametrization of the pre-images of rational space curves and rational surfaces under the mentioned stereographic projection. Differential-geometric properties of the considered curves and surfaces in the 3-sphere are also discussed.

Sava Grozdev, Veselin Nenkov (Bulgarian Academy of Sciences, Bulgaria) Two Pairs of Points, Generated by Central Conics with Respect to a Triangle The software program “THE GEOMETER’S SKETCHPAD” (GSP) is used in the paper to discover various interesting properties of special conics associated with a given ABC. For example, consider a point I in the plane of ABC and its conjugate IA IB IC with respect to ABC. The points I, IA , IB and IC are centres of conics k(I), k(IA ), k(IB ) and k(IC ), inscribed in ABC, while the mid-points of the segments IIA , IIB , IIC , IB IC , IC IA and IA IB lie ¯ on a conic k(O), which is circumscribed for ABC. Let the line I, parallel to IA IB , intersects CA and CB in points Ca and Cb respectively, while the line IC , parallel to IA IB , intersects CA and CB in points Ca and Cb respectively. Let Lc (I) = ACa ∩ BCb and Lc (IC ) = ACa ∩ BCb . Analogously, determine the points La (I), Lb (I), La (IA ) and Lb (IB ). Then, the lines ALa (I), BLb (I) and CLc (I) are concurrent with a point T (I). A variety of configurations and properties are considered in the paper too.

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Hans Havlicek (Vienna University of Technology, Austria) On M¨ obius Pairs of Simplices Two n-simplices in the n-dimensional projective space over a field F are mutually inscribed and circumscribed if each point of the first simplex is in a hyperplane of the second simplex, and vice versa for the points of the second simplex. Two such n-simplices will be called a M¨ obius pair of simplices or shortly a M¨ obius pair. The construction of M¨ obius pairs in any odd dimension is a straightforward task: Given any n-simplex take the images of its hyperplanes under any null polarity as vertices of a second simplex. By this approach, it remains open, though, whether or not the simplices have common vertices. We focus our attention to non-degenerate M¨ obius pairs. These are pairs of n-simplices such that each point of either simplex is incident with one and only one hyperplane of the other simplex. We present a construction of nondegenerate Möbius pairs which is based on null polarities. Furthermore, we exhibit the nested M¨ obius pairs arising from sub-simplices. Finally, we sketch how our results can be applied to construct distinguished systems of commuting/non-commuting operators in Pauli groups. This is joint work with Boris Odehnal and Metod Saniga.

Asen Hristov, Georgi Kostadinov (Plovdiv University, Bulgaria) On a Class of Almost Dual Manifolds The purpose of this note is to describe some properties of manifolds endowed with almost tangent (dual) structure F , F 2 = 0, ImF ⊂ KerF and almost complex structure J, J 2 = −id. Consider the algebra C ⊗ R(ε) with a basis {1, i, ε, k} and relations i2 = −1, iε = εi = k. It induces a G-structure on a manifold M, dim M = 4p + 2q. In any Tx M arise two operators F and K, F 2 = K 2 = 0, ImF = ImK = D1 , KerF = KerK = D1 ⊕ D2 , dim D1 = 2p, dim D2 = 2q and a (almost) complex structure J such that F J = JF = K. On T M we have two distributions < D1 > and < D2 > with dim D1 = 2p, dim D2 = 2q and J(D1 ) ⊂ D1 , J(D2 ) ⊂ D2 . The manifold M(F, J, K) with that structure we call the almost dual irregular manifold with almost complex structure. The simultaneous integrability conditions are: < D2 > is involutive, N (F, J) = N (F, F ) = N (J, J) = 0, where N is the Nijenhuis tensor for a couple of (1,1)-tensors. The distribution < D2 > defines a foliation F with typical fibre L, dim L = 2q. In the case NJ = N (J, J) = 0 we suppose that span{NJ (X, Y ) : X, Y ∈ Tx M} = D2 . Then M/F is a complex analytic manifold.

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Partially supported by project NI11-FMI-004 of the Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria.

Stefan Ivanov (Sofia University, Bulgaria) Heterotic Supersymmetry, Anomaly Cancellation and Equations of Motion The bosonic fields of the ten-dimensional supergravity of the form R1,9−d ×M d are the spacetime metric g, the NS three-form field strength H, the dilaton φ and the gauge connection A with curvature F A and are non-trivial only on M d , d ≤ 8. One considers the metric connection with totally skewsymmetric torsion ∇+ = ∇g + 12 H, where ∇g is the Levi-Civita connection of the Riemannian metric g. The heterotic equations of motion are 1 α A (F )imns (F A )mns − Rimns Rjmns = 0; Ricgij − Himn Hjmn + 2∇gi ∇gj φ − j 4 4 i −2φ ∇gi (e−2φ Hjk ) = 0; ∇+ (F A )ij ) = 0. i (e A heterotic geometry preserves supersymmetry iff there exists at least one Majorana-Weyl spinor such that the Killing-spinor equations hold:

∇m

+

= ∇ = 0;

1 dφ − H 2

· = 0;

F A · = 0,

(1)

where · means Clifford action of forms on spinors. The instanton equation, the last equation in (1) means that the curvature 2-form F A is contained in the Lie algebra of the Lie group which is the stabilizer of the spinor which in dimension 5,6,7 and 8 is the group SU (2), SU (3), G2 and Spin(7), respectively. The Green-Schwarz anomaly cancellation mechanism requires that the threeform Bianchi identity receives an α correction of the form dH =

α T r(R ∧ R) − T r(F A ∧ F A ) . 4

We show in [1] that the heterotic supersymmetry (Killing spinor equations) and the anomaly cancellation imply the heterotic equations of motion in dimensions five, six, seven, eight if and only if the connection on the tangent bundle is an instanton. For heterotic compactifications in dimension six this reduces the choice of that connection to the unique SU (3) instanton on a manifold with stable tangent bundle of degree zero. Reference: [1] S. Ivanov, Heterotic supersymmetry, anomaly cancellation and equations of motion, Phys. Lett. B 685, 190–19 (2010)

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Samet Karaibryamov, Bistra Tsareva, Boyan Zlatanov (Plovdiv University, Bulgaria) Educational Software for Interactive Training of Students on the Theme “Mutual Intersecting of Pyramids and Prisms in Axonometry” This work acquaints with a program SAM, written in “C .N ET ”, for interactive computer training of students on the theme “Mutual intersecting of pyramids and prisms in axonometry”. This program allows to follow the manual solution of the tasks, where the method of the auxiliary planes is used. It is enough to enter the coordinates of the vertices of the two polyhedra and our programme offers to the user a complete solution of the task in each of the five popular axonometric projection (cabinet projection, cavalier perspective, military perspective, increased orthogonal dimetry, increased orthogonal isometry). It includes the final sketch, a list of stages in the solution with a brief comments for the significant ones and a sequence of all steps in the solution. The stages in the solution of any task are: displaying of the bodies; constructing of the apex line and its steps; constructing consecutively of the pierce points of all surrounding edges; developing the pierce scheme; correcting the visibility of the edges conformed to the mutual intersecting of the bodies and hiding all auxiliary lines; presenting of the three-dimensional composition of both intersecting polyhedra in perspective. One can select the most suitable projection for the concrete task and by the buttons for presentation to consider the constructions in each stage with a possibility to move in both directions. Let denote that during the presentation all buildings, not participating in current and subsequent stages, are hidden. The user is free to experiment on the drawing manipulating its every subject separately (to change the style, colour, size) or to apply on the whole sketch rotation, translation, expansion, shrink, zoom, print, etc. A classification of the pierce points of the surrounding edges, based on the mutual position of their auxiliary planes, is made and as a result of the investigations we introduced the conceptions: tangency point, tangency segment and tangency area. All this allowed us to crawl all options of an intersection when the bases do not participate, to inform the program for them and to prepare it to respond adequately. Our programme and classification are instruments through which the user can compose easily and quickly not only standard tasks but and more interesting ones which miss in the known textbooks and teaching aids. One can change the coordinates of some vertices of the polyhedra in order to obtain improvisations on the given task, including the whole diversity, described by the classification. Our approach provides more in-depth studying of the topic and simultaneously facilitating its conditions of each stage (preparation of the teacher; teaching and learning; self-work of the students; examination) as much as possible. The authors are partially supported by project NI11-FMI-004 of the Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria.

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Helmut Karzel (TU M¨ unchen, Germany) Sayed-Ghahreman Taherian (Isfahan Univ. of Technology, Iran) Elliptic Reflection Spaces and Corresponding Kinematic Structures A reflection group is a pair (P, Γ) consisting of a group Γ and a non-empty subset P of the set J of all involutory elements of Γ such that: (S1) P generates the group Γ. Moreover if we set for a, b, x ∈ P , Π(a) := {x ∈ P \ {a} | xa = ax} ab := {x ∈ P | abx ∈ J} if ab = ba, ab := {x ∈ P \ Π(a) | abx ∈ J} ∪ {b} if ab = ba, a = b, and L := {ab | {a, b} ∈ P2 } we claim: (S2) (Three reflection axiom) ∀L ∈ L ∀x, y, z ∈ L xyz ∈ P. Let Γ+ = P · P := {ab | a, b ∈ P }, the map + κ : 2P → 2Γ ; X → X · X := {xy | x, y ∈ X}, is called kinematic embedding. Considering the elements of P as points and of L as lines, the pair (P, L) is an incidence space. (P, Γ) is called elliptic reflection space if (P, L) is a projective space. A subspace S of (P, L) is called reducible if ∀a, b, c, d ∈ S, ∃u, v ∈ S : abcd = uv. Results: 1. 2.

3.

The reducible subspaces of dim > 1 of an elliptic reflection space are exactly the planes. If (P, Γ) is an elliptic reflection plane then the kinematic embedding (Γ+ , F) with F := {κ(L) | L ∈ L} turns (P, Γ) in a kinematic space (Γ+ , ·, G) with G := {αF | α ∈ Γ+ , F ∈ F} where (Γ+ , G) is a 3-dimen : Γ+ → Γ+ ; ξ → α · ξ −1 · α sional projective space. If we set for α ∈ Γ+ , α + := { +, < Γ + >) is a 3-dimenand then Γ γ | γ ∈ Γ+ } then the pair (Γ sional elliptic reflection space. If (P, Γ) is an elliptic reflection space of any dimension then the kinematic embedding is a bijection between the set of all planes and the set of all kinematic spaces derived from (Γ+ , F) with F := {κ(L) | L ∈ L}.

Tamar Kasrashvili, Aleks Kirtadze (Georgian Technical University, Georgia) On Some Combinatorial Properties of Diophantine Sets in Euclidean Spaces Let X be a point-set (finite or infinite) in the n-dimensional Euclidian space Rn . We say that this X is a Diophantine set if the distance between any two

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points from X is a natural number. The most simple example of an infinite Diophantine set is the set all integer numbers in R. A well-known result states that any infinite Diophantine subset X of the Euclidean plane R2 is necessarily collinear, i.e., all points of X belong to a certain line. Notice also that the analogous proposition is not valid for finite Diophantine sets in the R2 . Properties of various discrete point systems are considered in many works. The following statements are true. Lemma. Let X be a Diophantine subset of an n-dimensional sphere of integer radius r > 2, where n ≥ 1. Then in the Euclidean space Rn+2 there exists a Diophantine set Y containing X and such that card (Y \ X) = 2. Theorem 1. For any natural number n ≥ 2 there are Diophantine sets which have arbitrary many points and do not lie in a hyperplane of Rn . Theorem 2. Let X be an infinite Diophantine set in the Euclidian space Rn , where n ≥ 1. Then all points of X are collinear. The proof of Theorem 2 is is essentially based on intersection of hyperboloids in the Euclidean space Rn . The main fact here is that the intersection of sufficiently many algebraic curves, which are in general position, always yields the empty set. Remark. The analogue of the above Theorem 2is not valid for infinite-dimenei √ (i ∈ N) from the classical sional vector spaces. Indeed, the set of vectors 2 infinite-dimensional Hilbert space l2 , where e1 , e2 , . . . , en , . . . are elements of an orthogonal basis from l2 , is a Diophantine set in l2 , and it is clear that these points are not collinear (moreover, they do not lie in any finite-dimensional subspace of l2 ).

Alexander Kharazishvili (A. Razmadze Mathematical Institute, Georgia) On Some Topologic-Geometrical Properties of External Bisectors of a Triangle Dealing with various problems in classical Euclidean geometry, sometimes rather delicate set-theoretical or topological techniques is needed. To use such methods, the so-called “completeness axiom” is necessary. However, sometimes certain weak forms of this axiom turn out to be sufficient. One of them is the statement that any cubic equation over the reals has at least one real root. Such weak forms of the completeness axiom, allow us to consider some topologicgeometrical properties of external bisectors of an arbitrary triangle [A, B, C]. As usual, we denote by la∗ the external bisector of [A, B, C] corresponding to the vertex A (an analogous notation is utilized for the other two vertices of [A, B, C]).

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Theorem 1. Let Δ be an equilateral triangle and let ε > 0. Then in the εneighborhoods of the vertices of Δ there exist points A, B, C respectively, such that, for the side lengths a, b, c of [A, B, C] we have a < c < b, but la∗ = lb ∗. Theorem 2. Suppose that the side lengths a, b, c of a triangle [A, B, C] satisfy the disjunction (c ≤ a and c ≤ b) or c ≥ a and c ≥ b and let la∗ = lb∗ . Then a = b, so [A, B, C] is an isosceles triangle. Theorem 3. For a triangle [A, B, C], the equalities la∗ = lb∗ = lc∗ imply a = b = c.

Ivan Landjev (New Bulgarian University, Bulgaria) The Packing Problem in Projective Hjelmslev Spaces Let R be a finite chain ring with |R| = q m and R/ rad R ∼ = Fq . Let Π = PHG(R Rn ) be the n-dimensional projective Hjelmslev space. A spread of Π of type λ = (λ1 , . . . , λn ), m = λ1 ≥ λ2 ≥ · · · ≥ λn ≥ 0, is a partition of the point set of Π in subspaces of type λ. In particular, an r-spread of Π is a partition of the point set of Π in r-dimensional Hjelmslev subspaces. An r-spread of Π is called regular if its image under the natural homomorphism η : R → R/ rad R is a multiple of an r-spread of PG(n, q). Similarly to the case of projective spaces over finite fields, r-spreads of Π do exist if and only if r + 1 divides n + 1. The known proofs of this result are constructive and produce regular spreads only. In this talk, we give the first examples of non-regular line spreads of the Hjelmslev geometries over the chain rings Z4 , Z2 [X]/(X 2 ), Z9 , Z3 [X]/(X 2 ). Furthermore, we consider the problem of partitioning the points of Π into subspaces that are not necessarily Hjelmslev subspaces. This is equivalent to the problem of partitioning the module R Rn into non-free submodules of fixed type that meet trivially. We prove that for spreads of subspaces of certain type the necessary divisibility conditions are not sufficient.

Mancho Manev, Miroslava Ivanova (Plovdiv University, Bulgaria) A Classification of the Torsion Tensors on Almost Contact Manifolds with B-Metric and Known Natural Connections In this work we consider linear connections on almost contact manifolds with B-metric (M, ϕ, ξ, η, g). We decompose the space of the torsion (0,3)- tensors of all linear connections on the considered manifolds in 11 orthogonal and invariant subspaces with respect to the action of the structural group. Thus, we give a classification of the connections on these manifolds with respect to the properties of their torsion tensor. Natural connections are a generalization

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of the Levi-Civita connections derived from the B-metric g and the associated B-metric g˜. A linear connection is called a natural connection on (M, ϕ, ξ, η, g) if the almost contact structure (ϕ, ξ, η) and both the B-metrics g and g˜ are parallel with respect to it. As an application of the obtained classification, we determine the classes of two known natural connections on (M, ϕ, ξ, η, g), namely, the canonical connection introduced in [M. Manev, K. Gribachev, Conformally invariant tensors on almost contact manifolds with B-metric. Serdica Math. J. 20 (1994), 133–147] and the so-called ϕKT -connection, i.e. the natural connection with totally skew-symmetric torsion introduced in [M. Manev, A connection with totally skew-symmetric torsion on almost contact manifolds with B-metric, arXiv:1001.3800]. Partially supported by project NI11-FMI-004 of the Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria.

Velichka Milousheva (Bulgarian Academy of Sciences, Bulgaria) Marginally Trapped Surfaces with Pointwise 1-Type Gauss Map in Minkowski 4-Space A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We use that the principal lines generated by the second fundamental form determine a geometric moving frame field at each point of such a surface. The derivative formulas for this frame field imply the existence of seven invariant functions. A theorem of G. Ganchev and V. Milousheva states that each marginally trapped surface is determined up to a motion by these seven invariant functions satisfying some natural conditions. We find that the Laplacian of the Gauss map of a marginally trapped surface is expressed by five of these invariant functions. Imposing the condition that the surface has pointwise 1-type Gauss map, we obtain that three of the invariants are zero. We give necessary and sufficient conditions for a marginally trapped surface to have pointwise 1-type Gauss map and find all marginally trapped surfaces with pointwise 1-type Gauss map. Our main result states that a marginally trapped surface is of pointwise 1-type Gauss map if and only if it has parallel mean curvature vector field.

Alexander Petkov, Stefan Ivanov(Sofia University, Bulgaria) HKT Manifolds with Holonomy SL(n; H) M. Verbitsky proved in [1] that a compact HKT manifold has holomorphically trivial canonical bundle exactly when the holonomy of the Obata connection is a subgroup of the special quaternionic linear group SL(n; H) which is one

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of the possible holonomy groups of a torsion-free linear connection in the Merkulov-Schwachh¨ ofer list. We show in [2] that on an HKT manifold the holonomy of the Obata connection is contained in SL(n; H) if and only if the Lee form is an exact one form. As an application, we obtain compact HKT manifolds with holomorphically trivial canonical bundle which are not balanced. A simple criterion for nonexistence of HKT metric on hypercomplex manifold is given in terms of the Ricci-type tensors of the Obata connection. References: [1] M. Verbitsky, Hyper complex manifolds with trivial canonical bundle and their holonomy, Moscow Seminar on Mathematical Physics. II, 203–211, Amer. Math. Soc. Transl. Ser. 2, 221 (2007). [2] S. Ivanov, A. Petkov, HKT manifolds with holonomy SL(n; H), to appear in Int. Math. Res. Notices, (IMRN), arXiv:1010.5052

Malgorzata Pra˙zmowska (University of Bialystok, Poland) Projective Realizability of Veronesians over Semilinear Spaces The notion of k-Veronesians (over semi-linear spaces) was introduced by Naumowicz & Pra˙zmowski. They generalized so called 2-Veronesians over a projective space considered by Tallini and Melone. Let D = S, L be an incidence structure. Definition 1. For an integer k ≥ 1 we define Vk (D) := yk (S), B, where yk (S)is the family of k-element multisets on S and the blocks B are the sets of type {ak−r e : a ∈ L} for L ∈ L, e ∈ yr (S), and 0 ≤ r ≤ k. Let distinct blocks of D have ≤ λ points in common; same holds for Vk (D). If D is a (v r bκ )-configuration then Vk (D) is

v+k−1 v+k−1 bκ . k kr k−1 Fact 1. (i) D ∼ = V1 (D). (ii) Vk (D) can be embedded in Vm−k (D) and Vk+m (D), where m is a positive integer. (iii) D can be embedded in Vk (D). (iv) If D is a substructure of D then Vk (D ) can be embedded in Vk (D). Definition 2. Combinatorial k-Veronesian is a structure Vk (S) = Vk (D), where D = S, {S}. We write Vk (n) = Vk (S) if |S| = n. Fact 2. Vk1 (n1 ) can be embedded in Vk2 (n2 ) if (and only if ) n1 ≤ n2 and k1 ≤ k2 . Fact 3. V3 (3), V2 (n) and Vn (2) are realizable on the real projective plane.

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Let n ≥ 3 or k ≥ 3. Then Vk (n) can be embedded in a Desarguesian projective space iff n = 3 = k or n = 2 or k = 2. If a Desarguesian projective space contains V3 (3) then the characteristics of its coordinate field is = 2. Theorem. Let k ≥ 3 and M be a semilinear space. If either M contains a line on > 3 points or k = 3 and M contains a line with at least 3 points then Vk (M) cannot be realized in a Desarguesian projective space. If the Fano configuration is contained in M (in particular, if M = P G(n, 2)) then V3 (M) is not realizable on any Desarguesian projective space. Conjecture. Let F be the projective Fano space. Then V2 (F) can be embedded in P G(4, 2).

Vencislav Radulov (University of Architecture, Civil Engineering and Geodesy, Bulgaria) Canonical Form of the Fundamental Matrix of a Bicentral Projection In the bicentral projection with projecting system (C, ω), (C , ω ) an important place takes the fundamental matrix F . This matrix matches an arbitrary point taken from w to an epipolar line from w . In the article all of the possibilities in the different cases for all mutual positioning of the two projecting planes and the baseline CC are explored. The presenting of the fundamental matrix with elements that are whole numbers and minimal 2-norm are exposed. Basis in both planes with which the canonical forms are realized are found. The eigenvalues of the fundamental matrix are {0; i; −i} or {0; 1; −1}.

Assia Rousseva, Ivan Landjev (Sofia University, Bulgaria) On the Existence of Some Optimal Arcs in P G(3, 5) The problem of finding the value of nq (k, d), defined as the minimum length of a q-ary linear code of dimension k and minimum distance d, is known as main problem in coding theory. It has been subject of intensive research in the past fifty years. This problem is usually tackled for small dimensions and small finite fields for all d. In the case of k = 4, q = 5, there are exactly four values of d for which n5 (4, d) is not known [Y. Edel, I. Landjev, On multiple caps in finite projective spaces, Designs, Codes and Cryptography 56 (2010), 163–175, T. Maruta, http://www.mi.s.oskafu-u.ac.jp/maruta/griesmer.htm]. They are given in the table below.

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d 81 82 161 162

g5 (4, d) 103 104 203 204

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n5 (4, d) 103–104 104–105 203–204 204–205

In this talk, we prove the non-existence of codes with parameters [104, 4, 82]5 . This implies that n5 (4, 82) = 105 thus settling one of the four open cases for k = 4, q = 5. Our approach to the problem is a geometric one. The existence of a code with parameters [104, 4, 82]5 is equivalent to that of a (104, 22)-arc in PG(3, 5). The restriction of a (104, 22)-arc to a maximal plane is a (22, 5)-arc. All arcs with these parameters are known, which sets restrictions on the structure of a hypothetical (104, 22)-arc. Our proof is completely computer-free.

Idzhad Sabitov (Moscow State University, Russia) Volume Polynomials for Some Polyhedra in n-Space It is known that for any simplicial polyhedron P in the Euclidean 3-space there exists a polynomial such that the volume of P is a root of the polynomial the coefficients of which being depending only on the lengths of edges of P and its combinatorial structure, see e.g. [I. Kh. Sabitov, Algebraic methods in the solution theory of polyhedra, Uspekhi Math. Nauk 66(3) (2011), 3–67]. The existence of such a polynomial for volumes of polyhedra in Euclidean n-spaces (n¿3) remains for many years as an open question. We are considering some classes of polyhedra in n-spaces for which one can prove the existence of the volume polynomials. Among these polyhedra we can mention following types of polyhedra: 1) 2) 3)

pyramides (that is polyhedra which have at least one vertex joined with all others vertices by edges); suspensions (or bipyramides) over pyramides; cross-polytopes.

In the same time we prove that these polyhedra are non-flexible if they are immersed. The work is supported by RFBR, grants No 09-01-00179 and No 10-0191000ANF.

Grozio Stanilov (Sofia University, Bulgaria) A Generalization of M¨ obius Strip We give generalizations of the classical Möbius strip. Namely we define the 8-parametrical family of surfaces:

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pu pu x0 = a cos(qu) + cv cos cos(qu), , y0 = b sin(qu) + dv cos sin(qu) , 2 2 pu z0 = k + ev sin 2 of parameters 0 ≤ u ≤ 2π, −1 ≤ v ≤ 1. One gets M¨ obius strip when a = 1, b = 1, c = 1, d = 1, e = 1, p = 1, q = 1, k = 0. For all these surfaces with exception of q = 0 the curve v = 0 is an ellipse. Then we prove the following: Theorem 1. For all values of a, b, c, d, e, k and arbitrary integers p, q with p odd number, the surface (x0 , y0 , z0 ) is non orientable. Theorem 2. For any number k the k-th degree of M¨ obius strip is a non-orientable surface. Theorem 3. The linear combination of the classical M¨ obius strip and the 3-fold M¨ obius strip is a non-orientable surface if the sum of the coefficients in the linear combination is non zero.

Ivaylo Staribratov, Penka Rangelova (Plovdiv University, Bulgaria) Experiments with Menelaus Theorem Menelaus theorem is associated with the co-linearity of three points which are incidental with corresponding three lines defined by the sides of a triangle. It is applicable in problem solving connected with the common point of three or more lines. If a, b and c are three lines, then the following two cases are possible: 1.

Consider two triangles with a common side on the line c. The line a has common points with the other two sides of the first triangle, and the line b has common points with the other two sides of the second triangle (Problem 1 and Problem 2).

2.

Consider a triangle with a side on the line c and the other lines intersecting the other two sides of the triangle (Problem 3).

Discussing the topic with students from 9th grade of OMG “Acad. Cyril Popov” in Plovdiv, a 2-group approach was applied—the students were divided into two groups. The first group was lectured in the conventional way while the second one was taught using the GeoNext software. All the students were equipped by computers and few examples were given to them. The approach was characterized by a high effectiveness. The choice of the proposed problems was not connected with standard textbooks which assured a different view of the Menelaus theorem examination. The experiment met its objectivity in the methodological approach which allowed a successful learning of a complex material in Geometry.

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Partially supported by project NI11-FMI-004 of the Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria.

Marta Teofilova (Plovdiv University, Bulgaria) Complex Connections on a Class Complex Manifolds with Norden Metric We introduce an eight-parametric family of complex connections, i.e. linear connections which preserve the complex structure by covariant differentiation on a basic class of complex manifolds with Norden metric, namely the class W1 of the classification of G. Ganchev and A. Borisov. We study the properties of the torsion tensors of these connections and obtain a four-parametric subfamily of symmetric complex connections. In particular, the well-known Yano connection is a member of this family. We also find necessary and sufficient conditions for the studied connections to be natural, i.e. to preserve both, the complex structure and the Norden metric. By this way we obtain a two-parametric family of natural connections on W1 -manifolds, in which the well-known canonical connection is contained. We study the curvature properties of the introduced complex connection on conformal K¨ ahler manifolds with Norden metric, i.e. W1 -manifolds which are conformally equivalent to K¨ ahler manifolds by the usual conformal transformation of the metric. We obtain necessary and sufficient conditions for their curvature tensors to be K¨ ahlerian and prove that the Bochner tensor of these connections is a conformal invariant. We also study natural connections whose curvature tensor is conformally invariant. The author is partially supported by project NI11-FMI-004 of the Scientific Research Fund, Paisii Hilendarski University of Plovdiv, Bulgaria.

Stoycho Trichkov (Sofia University, Bulgaria) Fluids Representation in Y n -spaces A generalized representation of fluid equations is adduced for Y n -spaces. Linearity and elasticity are surveyed by wielding various metrics enabling explicit elicitation. Specific cases in Y n -spaces are scrutinized.

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Veselin Videv (Trakia University, Bulgaria) Characterization of the Riemannian and Pseudo-Riemannian Manifold by Commuting Curvature Operators In the Riemannian and pseudo-Riemannian manifold using curvature tensor of the Manifolds and some curvature operators, we characterize these classes of manifolds, for which curvature operators commute at any point of the manifold.

Dirk Windelberg (Leibniz Universität Hannover, Germany) Hot Spot Within a conglomerate of residential buildings we are looking for a ranking with respect to annoyance and disturbance of railroad freight cars. The building with the highest impact is called a Hot Spot - and will get noise-rehabilitation first. I will look for possible mathematical definitions of a ranking-number which I will call “railway-noise-index RN I”. Let J be the number of railway noise levels pj (1 ≤ j ≤ J ) of a “typical” night. In the literature we find RN I = R ·

J

W (p(j) + D) ,

j=1

where D is the insulation of the wall and window between free space and inside space (in general D = −25 dB(A), that means “closed windows”). For a Hot Spot, we determine N RI (B(k)) for the buildings B(k) of the conglomerate K: Let us consider a building A with R(A) residents and with railway noise RN (A) =

J

W (p(j) + D) .

j=1

(1)

If B is another building with R(B) = 2 · R(A) residents and with the same railway noise, then RN I(B) = 2 · RN I(A).

(2)

If B is another building with R(B) = R(A) residents and with double noise, then RN (B) =

J

W (p(j) + 10 + D) and RN I(B) = RN I(B).

j=1

But what is a reasonable solution for RNI?