#### List of the talks and abstracts

• Acad. Radu MIRON

• Title: Some new Mechanics based on Lagrangian and Hamiltonian Geometries

• Mihai ANASTASIEI

• Title: Some generalizations of the Myers theorem

Abstract: The Myers theorem, appeared in the Riemannian geometry as a generalization of a Bonnet theorem on surfaces, extracts some topological properties of a Riemannian manifold from conditions on the Ricci scalar using the properties of geodesics, Jacobi fields and index form.
In this talk we present, in the Finslerian framework, two generalizations of the Myers theorem. These differ by conditions on the Ricci scalar as well as by the techniques of proof. The both extend some results from the Riemannian geometry published by Gregory J. Galloway around 1980.
In the first, the condition that the Ricci scalar is bounded below by a positive constant is replaced with a weaker one that allows negative values for it.
In the second it is assumed that the integral of the Ricci scalar on [0,∞) diverges to +∞ and one infers that the manifold is compact. Its hypothesis is weaker than of the first but it is not able to provide any information on the diameter of M. The proof of the first is standard while the proof of the second is based on a result regarding the solutions of a class of second order differential equations called of Jacobi type.

• Constantin ARCUŞ

• Title: Metrizability of the Lie algebroid generalized tangent bundle and (generalized) Lagrange (ro,eta)-spaces

Abstract: A class of metrizable vector bundles in the general framework of generalized Lie algebroids have been presented in the eight reference. Using a generalized Lie algebroid we obtain the Lie algebroid generalized tangent bundle of a vector bundle. This Lie algebroid is a new example of metrizable vector bundle. A new class (generalized) Lagrange spaces, called by use, (generalized) Lagrange (ro,eta)-spaces and Finsler (ro,eta)-spaces are presented. In the particular case of Lie algebroids, new and important results are presented. In particular, if all morphisms are identities morphisms, then the classical results are obtained.

• Vladimir BALAN

• Title: Tucker HO-SVD theory for 3-rd order Grobner-type Finsler structures

Abstract: The talk describes the Tucker HO-SVD theory, taylored for the 3-rd order Grobner locally Minkowski Finsler symmetric multilinear form of Berwald-Moor type. The k-modes and the different form approximants are discussed, and the geometrical background accompanying the specific tensor spectral algebra is highlighted.
The relations which occur while increasing the order and dimension in the Grobner form are investigated from multilinear spectral algebra point of view.

• Bogdan CÂNEPĂ

• Title: Algorithm for fixed point theorem for CM elliptic curves

• Elena Cristina CÂNEPĂ

• Title: Numerical simulations of defaults in large banking systems

• Monica CIOBANU and Virgil OBĂDEANU

• Title: Geometrical Structures Associated to Second Order Dynamical Systems

Abstract: The paper defines a geometric structure of the first and respectively second order, consisting of a d-one-form, a proper d-two-form (d-metric) and a d-connection, with which a geometry is constructed: by considering a covariant derivative (over the algebra of tensors fields), a parallel transport of vector fields, autoparallel curves, curvature tensors, etc. The geometric structure is identified with the derivation operator.
We consider first-order, respectively second order differential dynamical systems, and we show that any such system is associated with one of these geometric structures, such that the system's trajectories are autoparallel curves with respect to the associated structure.
We then consider non-holonomic dynamical systems and we show that they are in a bijective correspondence with the above geometric structures.

• Oana CONSTANTINESCU

• Title: Formal integrability for the nonautonomous case of the inverse problem of the calculus of variations

Abstract: We address the integrability conditions of the inverse problem of the calculus of variations for time-dependent SODE using the Spencer version of the Cartan-Kaehler Theorem. We consider a linear partial differential operator P given by the two Helmholtz conditions expressed in terms of semi-basic 1-forms and study its formal integrability. We prove that P is involutive and there is only one obstruction for the formal integrability of this operator. The obstruction is expressed in terms of the curvature tensor R of the induced nonlinear connection. We recover some of the classes of Lagrangian semisprays: flat semisprays, isotropic semisprays and arbitrary semisprays on 2-dimensional manifolds.

• Mircea CRÂŞMĂREANU

• Title: A deformation of conics and Finsler geometries

Abstract: The aim of this paper is to obtain new examples of Finsler structures having as model a scalar deformation of conics. Is a continuation of [1] from the point of view of relationship between quadratic polynomials (as equations of conics in dimension 2) and Finsler geometries.
References:
[1] O. Constantinescu, M. Crasmareanu, Examples of conics arising in two-dimensional Finsler and Lagrange geometries, An. Stiint. Univ. “Ovidius” Constanta Ser. Mat., 17(2)(2009), 45-59. MR2561008 (2010j:53031), Zbl 1199.53155

• Simona DRUŢĂ-ROMANIUC and Cornelia-Livia BEJAN

• Title: Harmonic connections and harmonic structures with respect to natural metrics

Abstract: We find all the general natural metrics on TM, with respect to which any (non)linear connection on a (pseudo)-Riemannian manifold (M,g) is harmonic. Moreover, we give necessary and sufficient conditions such that the Levi-Civita connection of g, the (non)linear connections on (M,g), and some special (1,1)-tensor fields on TM are harmonic with respect to any general natural metric G. We also study the harmonicity of these connections and of several (1,1)-tensor fields with respect to the natural diagonal metrics, and in particular, we obtain that the Levi-Civita connection of g is harmonic with respect to the Sasaki metric on TM, while the canonical almost tangent and almost complex structures on TM are harmonic with respect to the Sasaki metric on TM if and only if the base manifold is flat.

• Hiroshi ENDO and Shigeo FUEKI

• Title: The flag curvatures on projectivised tangent bundles deduced from contact metric structures constructed by certain Riemannian metric

Abstract: In References numbers [5] and [6], Sasaki type metric and more generalized Riemannian metric (h-v metric g~ [7]) were considered as a Riemannian metric constructing a contact metric structure deduced from the contact structure on the projectivised tangent bundle PTM. In this paper, we consider this h-v metric g~ under the certain conditions and study the flag curvature on PTM.

• Sinem GÜLER

• Title: Conformally Flat Special Generalized Quasi Einstein Spacetimes

Abstract: The object of the present paper is to study special generalized quasi Einstein spacetimes. The applications of this spacetime in general relativity are investigated. Among others, we prove Ricci semi symmetric generalized quasi Einstein spacetime is a nearly quasi Einstein spacetime and we show that Ricci semi symmetric generalized quasi Einstein spacetime satisfying Einstein's field equation with cosmological constant can be taken as a model of the perfect fluid spacetime in general relativity and cosmology. We also show that a conformally flat Ricci semi symmetric generalized quasi Einstein spacetime has a proper concircular vector field and this spacetime is a subprojective spacetime in the sense of Kagan. On the other hand, conformally flat Ricci semi symmetric generalized quasi Einstein spacetime can be expressed as a warped product Ix{eq}M* where M* is a 3-dimensional manifold of constant curvature and we also prove that this spacetime is the Robertson-Walker spacetime. Finally, we prove that this spacetime is a special conformally flat spacetime and every simple connected conformally flat Ricci semi symmetric generalized quasi Einstein spacetime can be considerably immersed in a Euclidean space E{n+1} as a hypersurface.

• Katica R. (STEVANOVIC) HEDRIH

• Title: Generalized function of fractional order disipation of system energy and extended Lagrange differential equation in matrix form

Abstract: A theory of vibrations of discrete fractional order system with finite number of degrees of freedom is founded in matrix form. A generalized function of visoelastic creep fractional order disipation of system energy and generalized forces of system no ideal visoelastic creep fractional order disipaion of system energy for generalized coordinates are introduced and deffined.
Extended Lagrange differential equations second order for fractional order system dynamics in matrix formformal form are introduced. Two theorems are formulated.
By use presented matrix method, as special case, the fractional order chain system dynamics is considered. Two body fractional order system dynamics in plane is considered.

• Cristina-Elena HREŢCANU and Mircea CRÂŞMĂREANU

• Title: On the properties of submanifolds in Riemannian manifolds with metallic structure

Abstract: The purpose of our paper is to give some properties of submanifolds endowed with a σ metallic Riemannian structure induced on a submanifold M by a metallic Riemannian structure (g,J) on a Riemannian manifold \overline{M} (i.e. J^2=pJ+qI, I is the identity operator on the Lie algebra \mathfrak{X}(\overline{M}) of the vector fields on \overline{M} and p,q are fixed positive integer numbers) and g is J-compatible (i.e. g(JX,Y)=g(X,JY) for every X,Y\in \mathfrak{X}(\overline{M})). A σ metallic Riemannian structure is a structure Σ =(P, g, u_{\alpha},\xi _{\alpha}, (a_{\alpha\beta})_{r}) induced on a submanifold M in a metallic Riemannian structure (\overline{M},g,J) determined by an (1,1) tensor field P on the submanifold M in \overline{M}, $\xi_{\alpha}\in \mathfrak{X}(M)$, $u_{\alpha}$ are 1-forms on M and $(a_{\alpha\beta})_{r}$ is an $r \times r$ matrix of smooth real functions on M (r is the codimension of M in \overline{M}). We search for the condition of normality of our structure similar to that for the almost paracontact structure.

• Toyoko KASHIWADA

• Title: On CR-submanifolds of l.c.K.-manifolds

Abstract: An l.c.K.-manifold is a Hermitian manifold whose metric is conformal to a Kaehler metric in local. As is known, such property is characterized by the existence of a closed 1-form $\alpha$ satisfying
$d\Omega = 2\alpha\wedge\Omega$,
with the fundamental 2-form $\Omega$.
In this talk, we discuss on CR-submanifolds in l.c.K.-manifolds considering the relation of $\alpha$ with the CR-distribution, and give some examples of CR-submanifolds.

• Hiroaki KAWAGUCHI

• Title: Consideration to the geometry of generalized metric spaces

Abstract: It is a historical fact that the differential geometry of curves and surfaces in 3-dimensional Euclidean space has developed the differential geometry on surfaces as 2-dimensional spaces with curvature independent of the ambient Euclidean space, which afterward has been generalized into the case of n-dimensional space as differential geometry of Riemannian spaces.
In Riemannian spaces the square of the fundamental metric function is given by the quadratic form of the differential dx at each point x of the space whose coefficients are functions of the position x only, while in Finsler spaces the coefficients are generalized into functions of not only the position x but also the direction dx, and while in Higher Order spaces the coefficients are more generalized into functions of x, dx, dx^2 …
The speaker and author is interested in the research of the natural or necessary extension or generalization of various geometry but less than in many researches of the formal extension or generalization.

• Bahar KIRIK and F.Ö. ZENGIN

• Title: On Almost Pseudo Ricci Symmetric Generalized Quasi-Einstein Manifolds

Abstract: In the present paper, we investigate almost pseudo Ricci symmetric generalized quasi-Einstein manifolds. Considering the properties of almost pseudo Ricci symmetric generalized quasi-Einstein manifolds, some special vector elds on these manifolds are examined. We prove some theorems about these vector fields.

• Koji MATSUMOTO

• Title: Semi-slant submanifolds with the symmetric ∇σ in a locally decomposable Riemannian manifold of (a,b)-type

Abstract: Recently, we considered the length of the second fundamental form and the mean curvature of a semi-slant submanifold in a locally decomposable Riemannian manifold with an almost constant curvature and gave some inequalities of these ([4]). In this talk, we consider a semi-slant submanifold with the symmetric ∇σ in a locally decomposable Riemannian manifold with an almost constant curvature and we prove that a locally decomposable Riemannian manifold of (a,b)-type is locally Euclidean if and only if its semi-slant submanifold has the symmetric ∇σ.
References:
[1] B.Y, Chen, Geometry of submanifolds, Marcel Dekker, New York (1973).
[2] B. Kirik and F.Ö. Zengin, On nearly quasi-Einstein manifolds, to appear.
[3] K. Matsumoto, On submanifolds of locally product Riemannian manifolds, TRU Math., 18-2 (1982), 145--157.
[4] K. Matsumoto and Z. Senturk, Submanifolds in locally decomposable Riemannian manifolds, to appear.
[5] B. Sahin, Warped product semi-slant submanifolds of locally product Riemannian manifolds, Bull. Math.Soc. Sci. Math. Roumanie, 49, (2006), 383--394.
[6] B. Sahin, Warped product semi-slant submanifolds of locally product Riemannian manifolds, Studia Sci. Math. Hungarica 46 (2) (2009), 169--184.
[7] K. Yano, Differential Geometry on complex and almost complex spaces, Pargamon Press, (1965).

• Roman MATSYUK

• Title: Covariant Second Order Mechanics in the Riemannian Space

Abstract: In a (pseudo)Riemannian manifold we give a covariant and coordinate-homogeneous description of the second order Ostrohrads'kyj mechanics, in the case where the integral curves are invariant under the changes of the dynamical parameter along them. Following Grässer, Rund, and Weyssenhoff, we present, in terms of covariant derivatives, the system of the generalized canonical equations for the second order autonomous and parameter-ambivalent variational problem in the form of an exterior differential system with the Zermelo constraint.

• Adela MIHAI and Ion MIHAI

• Title: Certain Distinguished Vector Fields on Riemannian Manifolds

Abstract: We establish relationships between the concepts of a torse forming vector field and of exterior concurrent and quasi-exterior concurrent vector fields. It is proved that any torse forming is a quasi-exterior concurrent vector field. We obtain a necessary and sufficient condition for a torse forming vector field to be 2-exterior concurrent and give a foliation. We provide applications of the existence of torse formings on Sasakian manifolds and Kenmotsu manifolds.
References:
[1] A. Mihai and I. Mihai, Torse forming vector fields and exterior concurrent vector fields on Riemannian manifolds and applications, Journal of Geometry anf Physics 73 (2013), 200-208.
[2] I. Mihai, R. Rosca and L. Verstraelen, Some aspects of the Differential Geometry of Vector Fields, PADGE 2, K. U. Brussel, K. U. Leuven 1996.

• Gheorghe MUNTEANU and Nicoleta ALDEA

• Title: On the geometry of complex Cartan-Randers metrics

• Marian Ioan MUNTEANU

• Title: Reduction results for magnetic trajectories in Sasakian and cosymplectic manifolds

Abstract: We investigate the trajectories of charged particles moving in a space modeled by the 3-space M^2(c)xR under the action of the Killing magnetic fields. One explicitly determines all magnetic curves corresponding to the Killing magnetic fields on the 3-dimensional Euclidean space (c=0). See [1]. We give the local description of the magnetic trajectories associated to Killing vector fields in S^2xR, providing their complete classification (c=1). Moreover, some interpretations in terms of geometric properties are given. See [2]. Then, the geometry of normal magnetic curves in a Sasakian (respectively cosymplectic) manifold of arbitrary dimension is explained. Some results about the reduction of the codimension of a normal magnetic curve in a Sasakian space form are given. See [3].

References:
[1] S.L. Druta-Romaniuc, M.I. Munteanu, Magnetic curves corresponding to Killing magnetic fields in E^3, J. Math. Phys. 52(11) (2011), art. no. 113506.
[2] M.I. Munteanu, A.I. Nistor, The classification of Killing magnetic curves in S^2xR, J. Geom. Phys. 62(2) (2012), 170-182.
[3] S.L. Druta-Romaniuc, J. Inoguchi, M.I. Munteanu, A.I. Nistor, Magnetic curves in Sasakian and cosymplectic manifolds, submitted.

• Tetsuya NAGANO

• Title: Notes on the first and the second variations of Arc Length

Abstract: The author has been studying about "linear parallel displacement" in Finsler geometry from 2008 [1]. Last year the author was able to get three quantities (0.1), (0.2) and (0.3) supposed that he was concerned with a curvature of the space by investigating linear parallel displacements along a infinitesimal parallelogram [4]. In this year, the author studies the first variation and the second variation of Arc Length under the standpoint of linear parallel displacement, where this is different from the traditional manner. As a result, the author notices that three quantities mentioned above appeared in the second variation. By my lecture of today, I talk about it.

References:
[1] T. Nagano, Notes on the notion of the parallel displacement in Finsler geometry, Tensor N.S., 70(2008), 302-310.
[2] T. Nagano, A note on linear parallel displacements in Finsler geometry, Journal of the Faculty of Global Communication, University of Nagasaki, 12(2011), 196-205.
[3] T. Nagano, On the conditions for Finsler spaces to be locally at Riemannian spaces, 2011, preprint.
[4] T. Nagano, The quantities derived from linear parallel displacements along an infinitesimal parallelogram, Journal of the Faculty of Global Communication, University of Nagasaki, 13(2013), 129-141.

• Ana Irina NISTOR

• Title: On the geometry of constant angle surfaces

Abstract: In this talk we present some results obtained in collaboration with F.Dillen, Y.Fu and M.I.Munteanu in the study of constant angle property for surfaces in product spaces. First results consist in the classification of constant angle surfaces in M^2xR_1. Next, we study the surfaces endowed with a canonical principal direction in M^2xR_1 and H^2xR. Finally we classify all the surfaces making constant angle with a Killing vector field in the Euclidean 3-space.

• Mileva PRVANOVIC

• Title: Holomorphically Projective Mappings onto Semisymmetric Anti-Kaehler Manifolds

Abstract: We prove that if it is possible to H-projectively map anti-Kaehler manifold (M,g,J) on semisymmetric anti-Kaehler manifold (\bar{M}, \bar{g},J) then both manifolds are of constant totally real sectional curvature, or the manifold (M,g,J) is holomorphically pseudosymmetric.

• Marian RĂILEANU

• Title: A new interpretation of the Bragg's law

Abstract: The subject of the paper is about the diffraction of the X-ray in the radioactive materials.

• Adrian SANDOVICI and Marcel ROMAN

• Title: A class of Generalized Dirac Structures on Banach Lie Algebroids

Abstract: It is well known that a Dirac structure is as a sub--bundle of the “big” bundle TM⊕T*M that is equal to its orthogonal complement with respect to the symmetric Courant operator. Dirac structures were generalized to Banach Dirac bundles in [2]. In this paper a class of Banach Dirac bundles are introduced within the framework of infinite-dimensional Banach manifolds and their basic properties are derived. Banach Lie algebroids are geometrical objects recently introduced by Mihai Anastasiei in [1]. It is the main goal of the present paper to reveal the most important links between the introduced new class of Banach Dirac bundles and Banach Lie algebroids, respectively. All considerations are done in the category of Banach vector bundles, see [1,2,3].
References:
[1] M. Anastasiei, Banach Lie Algebroids, An. St. Univ. “Al.I. Cuza” Iasi S.N. Matematica, T. LVII, 2011 f.2, 409-416.
[2] M. Anastasiei and A. Sandovici, Banach Dirac Bundles Int. J. Geom. Methods Mod. Phys. 10, 1350033 (2013) [16 pages] DOI: 10.1142/S0219887813500333.
[3] S. Lang, Fundamentals of Differential Geometry, Graduate Texts in Mathematics 191, Springer, 1999.

• Petre STAVRE and Adrian LUPU

• Title: About some symplectic conjugations and almost symplectic conjugations

Abstract: In this paper the authors generalize some results regarding the ω-conjugation of the linear connections, obtained in the papers published in the previous numbers of the Tensor journal.
This generalizations are related to the invariants which implies the curvature tensor and the Bianchi Tensor, as well as the transformations of the connections pairs which preserve the ω-conjugation, respectively the absolute ω-conjugation.

• WALEED Ahmed Elsayed Mohamed

• Title: A Global Approach to Absolute Parallelism Geometry

Abstract: In this paper we provide a global investigation of the geometry of parallelizable manifolds(or absolute parallelism geometry) frequently used for application. We discuss the different linear connections and curvature tensors from a global point of view. We give an existence and uniqueness theorem for a remarkable linear connection, called the canonical connection. Different curvature tensors are expressed in a compact form in terms of the torsion tensor of the canonical connection only. Using the Bianchi identities, some interesting identities are derived. An important special fourth order tensor, which we refer to as Wanas tensor, is globally defined and investigated. Finally a “double-view” for the fundamental geometric objects of an absolute parallelism space is established: The expressions of these geometric objects are computed in the parallelization basis and are compared with the corresponding local expressions in the natural basis. Physical aspects of some geometric objects considered are pointed out.

• Violeta ZALUŢCHI

• Title: Harmonic maps between two second order holomorphic jets bundle