g-Steiner, co-Steiner and Normal Points of Bounded Euclidean Submanifolds

Yıl 2020, Cilt: 13 Sayı: 2, 1 - 10, 15.10.2020

Bang-yen Chen

https://doi.org/10.36890/iejg.689803

Öz

The original ``Steiner point'', also known as the ``Steiner curvature centroid'', is the geometric centroid of the system obtained by placing a mass equal to the magnitude of the exterior angle at each vertex of a triangle. Steiner points have been studied and applied in networks, combinatorics, computational geometry and even in game theory.

In this article, we extend the notion of Steiner point to the notion of g-Steiner point for a bounded Euclidean submanifolds with arbitrary codimension. In this article, we also introduce the notions of co-Steiner and normal points for bounded Euclidean submanifolds. We prove several basic properties for such points. Furthermore, we establish some links between g-Steiner, co-Steiner and normal points.

Anahtar Kelimeler

Steiner point, g-Steiner point, co-Steiner point, normal point, G-total curvature, Gauss-Kronecker curvature

Kaynakça

• \bibitem{Chen67} Chen, B.-Y., On the total absolute curvature of manifolds immersed in Riemannian manifold. {\it Kodai Math. Sem. Rep.} {\bf 19} (1967), 299--311.
• \bibitem{Chen68.1} Chen, B.-Y., Some integral formulas of the Gauss-Kronecker curvature. {\it Kodai Math. Sem. Rep.} {\bf 20} (1968), 410--413.
• \bibitem{Chen68.2} Chen, B.-Y., Notes on the G-Gauss-Kronecker curvature. {\it Nanta Math}. {\bf 2} (1968), 47--53.
• \bibitem{Thesis} Chen, B.-Y., On the G-total curvature and topology of immersed manifolds. Thesis (Ph.D.)-University of Notre Dame. 1970.
• \bibitem{Chen71.1} Chen, B.-Y., On the total curvature of immersed manifolds. I. An inequality of Fenchel-Borsuk-Willmore. {\it Amer. J. Math.} {\bf 93} (1971), 148--162.
• \bibitem{Chen71.2} Chen, B.-Y., On an integral formula of Gauss-Bonnet-Grotemeyer. {\it Proc. Amer. Math. Soc.} {\bf 28} (1971), 208--212.
• \bibitem{Chen72} Chen, B.-Y., G-total curvature of immersed manifolds. {\it J. Differential Geometry} {\bf 7} (1972), 371--391.
• \bibitem{Chen11} Chen, B.-Y., Pseudo-Riemannian manifolds, $\delta$-invariants and applications. World Scientific Publishing, Hackensack, NJ, 2011.
• \bibitem{Chen19} Chen, B.-Y., {Geometry of Submanifolds}; 2nd Edition, Dover Publications, Mineola, New York 2019.
• \bibitem{Chern57} Chern, S.-S. and Lashof, R. K., On the total curvature of immersed manifolds. {\it Amer. J. Math.} {\bf 79} (1957), 306--318.
• \bibitem{F66} Flanders, H., The Steiner point of a closed hypersurface. {\it Mathematika} {\bf 13} (1966), 181--188.
• \bibitem{KN12} Kamma, L. and Nutov, Z., Approximating survivable networks with minimum number of Steiner points. {\it Networks} {\bf 60} (2012), no. 4, 245--252.
• \bibitem{M70} Meyer, W. J., Characterization of the Steiner point. {\it Pacific J. Math.} {\bf 35} (1970), 717--725.
• \bibitem{O66} Otsuki T., On the total curvature of surfaces in Euclidean spaces. {\it Japanese J. Math.} {\bf 35} (1966), 61--71.
• \bibitem{P02} Pechersky, S., The Steiner point of a convex set and the cooperative games solutions. ICM2002GTA (Qingdao), 637--641, Qingdao Publ. House, Qingdao, 2002.
• \bibitem{S71} Schneider, R., On Steiner points of convex bodies. {\it Israel J. Math.} {\bf 9} (1971), 241--249.
• \bibitem{S66} Shephard, G. C., The Steiner point of a convex polytope. {\it Canadian J. Math.} {\bf 18} (1966), 1294--1300.
• \bibitem{S68} Shephard, G. C., A uniqueness theorem for the Steiner point of a convex region. {\it J. London Math. Soc.} {\bf 43} (1968), 439--444.
• \bibitem{S27} Su, B., On Steiner's curvature-centroid. {\it Japanese J. Math.} {\bf 4} (1927), 195--201.
• \bibitem{S1838} Steiner, J., Von dem Krummungsschwerpunkt ebener Kurven. {\it J. Reine Angew. Math.} (Crelle's Journal) {\bf 21} (1838), 101--133.