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Abstract  

It is proved that non-degenerate surfaces in R 4 with planar geodesics are only the complex paraboloid and the product of two parabolas.

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Abstract  

In this work a mistake in the paper is corrected. There is also a new proof of the main theorem which classifies the non-degenerate affine surfaces in R 4 having planar geodesics with respect to the affine metric.

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. , Problem 12b, Discrete Geometry Tagungsbericht 20 , Math. Forschungsinstititut Oberwolfach ( 1997 ). [2] Busemann, H” The geometry of geodesics, Academic Press (New York, 1955 ). MR IT,779a

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Summary  

We study minimal and totally geodesic submanifolds in Lie groups and related problems. We show that: (1) The imbedding of the Grassmann manifold G F (n,N) in the Lie group G F (N) defined naturally makes G F (n,N) a totally geodesic submanifold; (2) The imbedding S 7SO(8) defined by octonians makes S 7a totally geodesic submanifold inSO(8); (3) The natural inclusion of the Lie group G F (N) in the sphere ScN^2-1(√N) of gl(N,F)is minimal. Therefore the natural imbedding G F (N)→gl(N,F )is formed by the eigenfunctions of the Laplacian on G F (N).

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The geodesic graph of Riemannian spaces all geodesics of which are orbits of 1-parameter isometry groups was constructed by J. Szenthe in 1976 and it became a basic tool for studying such spaces, called g.o.\ spaces. This infinitesimal structure corresponds to the reductive complement \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\mathfrak m$ \end{document} in the case of naturally reductive spaces. The systematic study of Riemannian g.o. spaces was started by O. Kowalski and L.~Vanhecke in 1991, when they introduced the most important definitions, classified the low-dimensional examples and described the basic constructions of this theory. The aim of this paper is to investigate a connection theoretical analogue of the concept of the geodesic graph.

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Summary  

In this note we use the Hopf map π: S 3S 2 to construct an interesting family of Riemannian metrics h fon the 3-sphere, which are parametrized on the space of positive smooth functions f on the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. The Hopf tube over an immersed curve γ in S 2 is the complete lift π-1(γ) of γ, and we prove that any geodesic of this Hopf tube satisfies a Clairaut relation. A Hopf tube plays the role in S 3 of the surfaces of revolution in R 3. Furthermore, we show a geometric integration method of the Frenet equations for curves in those non-standard S 3. Finally, if we consider the sphere S 3 equipped with a family h f of Lorentzian metrics, then a new Clairautrelation is also obtained for timelike geodesics of the Lorentzian Hopf tube, and a geometric integration method for curves is still possible.

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Abstract  

We prove that the Hopf vector field is unique among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As an application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation using standard approach from the theory of submanifolds and find exact boundaries for the sectional curvature of the Hopf vector field.

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Abstract

The increasing cooperation in science, which has led to larger co-authorship networks, requires the application of new methods of analysis of social networks in bibliographic co-authorship networks as well as in networks visible on the Web. In this context, a number of interesting papers on the “Erdős Number”, which gives the shortest path (geodesic distance) between an author and the well-known Hungarian mathematician Erdős in a co-authorship network, have been published recently. This paper develops new methods concerning the position of highly productive authors in the network. Thus a relationship of distribution of these authors among the clusters in the co-authorship network could be proved to be dependent upon the size of the clusters. Highly productive authors have, on average, low geodesic distances and thus shorter length of paths to all the other authors of a specialism compared to low productive authors, whereas the influencing possibility of highly productive scientists gets distributed amongst others in the development of the specialism. A theory on the stratification in science with respect to the over random similarity of scientists collaborating with one another, previously covered with other empirical methods, could also be confirmed by the application of geodesic distances. The paper proposes that the newly developed methodology may also be applied to visible networks in future studies on the Web. Further investigation is warranted into whether co-authorship and web networks have similar structures with regards to author productivity and geodesic distances.

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In this paper geodesically corresponding metricsg and on a manifoldM, dim ≥5, under the assumption that the tensorsR andS of the metricg satisfyR.R=Q(S, R), are considered. It is stated that the corresponding tensors and of not necessarily must satisfy . Certain relations between the curvatures ofg and are obtained.

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Summary  

The discrete isoperimetric problem is to determine the maximal area polygon with at most \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $k$ \end{document} vertices and of a given perimeter. It is a classical fact that the unique optimal polygon on the Euclidean plane is the regular one. The same statement for the hyperbolic plane was proved by K\'aroly Bezdek [1] and on the sphere by L\'aszl\'o Fejes T\'oth [3]. In the present paper we extend the discrete isoperimetric inequality for ``polygons'' on the three planes of constant curvature bounded by arcs of a given constant geodesic curvature.

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