Δελτίο της Ελληνικής Μαθηματικής Εταιρίας: Τεύχος 44

For a Riemannian manifold (M, g), we show how the scalar curvature of a tangent sphere bundle T, M (endowed with the induced Sasaki metric) depends on the constant spherical radius r > 0. A special attention is paid to the case when the above scalar curvature is constant along each.

One of the typical examples of CR submanifold of maximal CR dimension of complex manifold is a real hypersurface. Therefore, we may expect to generalize some theorems which are valid on real hypersurface to theorems on CR submanifold of maximal CR dimension. In this paper, we state generalization of two theorems on real hypersurface. One is a generalization of a theorem of Cecil ? Ryan [1] and another is that of a theorem of Lawson [4]

This article presents a short overview and some new results on the theory of ray systems. (This paper is based on a talk presented by the author in the 4th Panellenic Conference on Geometry on Research in Geometry and in its Teaching towards the 21st century taking place in the University of Patras, Greece, 28-30 May 1999.)

The Weil ? Kostant theorem characterizes those alternating (real ? valued) 2 ? forms which are curvature forms of connections in U (1) ? bundles. We present in this talk an overview of a corresponding result for arbitrary Lie groups which was proved by K.C.H. Mackenzie in 1987, using the notion of Lie algebroids. A Lie algebroid is a vector bundle whose module of sections has a Lie algebra bracket and a vector bundle morphism to the tangent space of the base manifold which preserves the Lie brackets. The main aims of this talk are two: First, to demonstrate that the theory of Lie algebroids is a suitable environment in which one can do connection theory (which is this speaker?s main research interest), and second to show how one can use Lie algebroids to tackle non ? abelian problems which often arise in the process of quantization and elsewhere.

In this paper we study the shape operator of a tube about a certain submanifold P of a K?hler manifold M of constant holomorphic sectional curvature. We give a necessary condition for the manifold M to have constant holomorphic sectional curvature.

In this paper we examine the A ? range and the φ ? spatial numerical range of an elementary tensor in l.m.c. algebras and l.m.c. * - algebras, respectively. We find the relationship between the product of the A ? ranges (resp. φ ? spatial numerical ranges) of two elements and that of the A ? range (resp. φ ? spatial numerical range) of their corresponding tensor. On the other hand, we give sufficient conditions, under which this relation becomes equality.

The aim of the present paper is to examine the symmetry groups of an autonomous Hamiltonian system. The authors use the classical method of finding point transformations. According to their approach a complete classification for one and two degrees of freedom is presented as well as partial results for the case of three degrees of freedom. In general it is obtained a maximal dimension for the harmonic oscillator or a free particle while the other dimensions vary between 1 and n^2+3. Finally by considering velocity-dependent potentials linear in the momenta it is proved an interesting result concerning the symmetry group of the corresponding Hamiltonian system. (This paper is based on a talk presented by the author in the 4th Panellenic Conference on Geometry on "Research in Geometry and in its Teaching towards the 21st century'' taking place in the University of Patras, Greece, 28--30 May 1999.)

The aim of the present paper is to examine partitions of the spectrum mu(E) of a given topological algebra E, by means of suitable subsets of the spectrum by taking an appropriate covering of mu(E) and by considering the geometric hulls of its members. (The original result is given in Theorem 2.1). Furthermore according to author's approach the abovementioned decomposition of mu(E) may give a similar decomposition of the Gel'fand transform algebra of E by transforming it to a homeomorphism of the spaces involved within the addition of suitable conditions on the topological algebra, the spectrum and the initial covering. (The original results are given in Corollaries 2.1, 2.2). (This paper is based on a talk presented by the author in the 4th Panellenic Conference on Geometry on "Research in Geometry and in its Teaching towards the 21st century'' taking place in the University of Patras, Greece, 28--30 May 1999.)

The aim of the present paper is to study the resulting normal to the orbits distribution by considering isometric actions on Riemannian manifolds. In particular in this paper the author gives some new fibring results by relating basic connections to locally splitting actions. As an application it is described the 3-manifolds admitting a specific locally splitting action of codimension 1. (This paper is based on a talk presented by the author in the 4th Panellenic Conference on Geometry on "Research in Geometry and in its Teaching towards the 21st century'' taking place in the University of Patras, Greece, 28--30 May 1999.)

Let (A, M) and p: M|rightarrow M be an arbitrary Lie groupoid and submersion with connected fibers respectively. The aim of the present paper is to define a more general class of differential equations formulated in terms of the theory of groupoids and algebroids. For this purpose the quotient-groupoid (A/p, M) for every Lie groupoid (A, M) and every submersion p is constructed since these equations are defined on Lie algebroids of such groupoids. In particular, it is proved that (A/p, M) is a C^infty-subgroupoid of (A, M). Regarding this result a more general class of differential equations with generalized total differential are defined on the Lie algebroid L(A/p) with values in the Lie algebroid of a Lie groupoid (|Omega, ?). Moreover some specific examples of this approach along with a triviality criterion for groupoids are given such as the differential equations with total / logarithmic differential and values in Lie algebras in the case that A=M|times M and B={pt}. (This paper is based on a talk presented by the author in the 4th Panellenic Conference on Geometry on "Research in Geometry and in its Teaching towards the 21st century'' taking place in the University of Patras, Greece, 28-30 May 1999.)

The aim of the present paper is to define morphisms of differential triads so that the abstract differential structure is preserved. Following those results it is proved that differential triads and their morphisms form a category in which differential manifolds are embedded. Then it is proved that every subset of the base space of a differential triad determines a differential triad which is a subobject of the former, a property missing in the category of manifolds as well as that the category of differential triads haw finite products. (This paper is based on a talk presented by the author in the 4th Panellenic Conference on Geometry on "Research in Geometry and in its Teaching towards the 21st century'' taking place in the University of Patras, Greece, 28--30 May 1999.)

A *- vector space is a pair (V, *) consisting of a vector space V and a map *: V ? V: x a*x, x ? V, with the properties: **x = x, *()xy+ = **xy+, ** (λ x) = λ x , x, y V, λ C (linear involution or vector space involution). A * - algebra (or involutive algebra) is a * - vector space (A, *) whose involution satisfies the additional property: ???***(),,xyyxxy?=??V (algebraic involution of the algebra A). Here the following questions arise: Question 1: Can we find all (: linear) involutions of a given vector space V? Question 2: Can we find all (algebraic) involutions of a given algebra A? Question 3: Can we find all multiplications of a vector space V and, in particular, the associative ones? In this paper we answer in the positive the first and third question, while we give a relevant hint (cf. Theorem 1.3), concerning the second one.

The present paper covers a topic placed in the framework of Abstract Differential Geometry expound in the book of A. Mallios {|it ``Geometry of Vector Sheaves''}(Vols. I-II, Kluwer, Dordrecht, 1998) combined also with the work presented in a previous paper of the author ({|it "Connections on principal sheaves''} (New Developments in Differential Geometry, Budapest 1996 (J. Szenthe Ed. ), Kluwer, 459--483, 1999). Following the abovementioned framework the author studies sheaves associated with a given principal sheaf by means of morphisms between certain sheaves of groups called {it "Lie sheaves of groups''}. Furthermore according to author's approach it is examined the connections induced on the associate sheaves and their relation with the given one. (This paper is based on a talk presented by the author in the 4th Panellenic Conference on Geometry on "Research in Geometry and in its Teaching towards the 21st century'' taking place in the University of Patras, Greece, 28-30 May 1999.)

The aim of the present paper is to describe an efficient numerical method for locating and computing solutions of systems of nonlinear algebraic and transcendental equations as well as the relationship between this method and the Sperner lemma. In particular, the author following the idea of using bisection based methods in problems concerning numerical simulations gives a generalized bisection method applied on n-dimensional simplexes by avoiding constructed Sperner simplices even though the structure of method is based on the existence of a Sperner simplex. Furthermore, it is proved that the above method converges rapidly to a solution of large and imprecise problems since the only computable information required is the algebraic signs of the components of the function. (This paper is based on a talk presented by the author in the 4th Panellenic Conference on Geometry on ``Research in Geometry and in its Teaching towards the 21st century'' taking place in the University of Patras, Greece, 28--30 May 1999.)