Acta Mathematica Academiae Paedagogicae Nyíregyháziensis, Vol. 26, No. 2, pp. 359-375 (2010)

Examples and notes on generalized conics and their applications

Á. Nagy and Cs. Vincze

University of Debrecen

Abstract: Let $\Gamma$ be a subset of the Euclidean coordinate space. A generalized conic is a set of points with the same average distance from the points $\gamma \in \Gamma$. First of all we consider some realizations of this concept. Basic properties will be given together with an application. It is a general process to construct convex bodies which are invariant under a fixed subgroup $G$ of the orthogonal group in $\mathbb{R}^n$. Such a body induces a Minkowski functional with the elements of $G$ in the linear isometry group. To take the next step consider $\mathbb{R}^n$ as the tangent space at a point of a connected Riemannian manifold $M$ and $G$ as the holonomy group. By the help of the method presented here $M$ can be changed into a non-Riemannian Berwald manifold with the same canonical linear connection as that of $M$ as a Riemannian manifold. Indicatrices with respect to the Finslerian fundamental function are generalized conics with respect to the Euclidean norm induced by the Riemannian metric.

Keywords: Minkowski functionals, Finsler manifolds, Berwald manifolds

Classification (MSC2000): 53C60; 58B20

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