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研究生： 黃志強
論文名稱： 在二維黎曼流形上的四頂點定理
The Four-vertex Theorem in Riemannian 2-manifolds
指導教授： 林俊吉
學位類別： 碩士
Master
系所名稱： 數學系
Department of Mathematics
論文出版年： 2006
畢業學年度： 94
語文別： 英文
論文頁數： 15
中文關鍵詞： 頂點 、曲率 、外接圓 、四頂點定理
英文關鍵詞： vertex, curvature, circumscribed circle, four-vertex theorem
論文種類： 學術論文
相關次數： 點閱：177  下載：6 
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    • 我們研究從1909年開始的頂點定理之結果。更進一步，我們著重於在二維黎曼流形上的曲線，並且推廣奧瑟曼的證明。在這篇論文，某些二維黎曼流形上特殊曲線的四頂點定理被得到。

    We survey the results on the four-vertex theorems, beginning from 1909. Moreover, we focus on curves in Riemannian 2-manifolds and generalize the Osserman’s proof [6]. Some four-vertex theorems for special curves in Riemannian 2-manifolds are derived in this thesis.

    §1 Introduction 1 §2 Comparisons and discussions on the proofs of the four-vertex theorem 1~2 §3 The four-vertex theorem in Hadamard’s 2-manifolds with negative constant curvature 2~5 §4 The four-vertex theorem in Hadamard’s 2-manifolds for some special curves 5~11 §5 The four-vertex theorem in Riemannian 2-manifolds for some special curves 11~13 §6 Summary 13~14 Reference 15

    [1] S. B. Jackson, The four-vertex theorem for spherical curves, Amer. J. of Math. 62 (1) (1940) 795-812.

    [2] S. B. Jackson, Vertices of plane curves, Bull. A.M.S. 50 (1944), 564-578.

    [3] S. B. Jackson, The four-vertex theorem for surfaces of constant curvature, Amer. J. Math. 67 (1945), 563-582.

    [4] S. B. Jackson, Geodesic vertices on surfaces of constant curvature, American Journal of Mathematics 72(1), (1950), 161-186.

    [5] G. Thorbergson, Vierscheitelatz auf Flaechen nichtpositiver Kruemmung, Math. Z. 149 (1976), 47-56.

    [6] R. Osserman, The four-or-more vertex theorem, Amer. Math. Monthly 92 (1985), 331-337.

    [7] David A. Singer, Diffeomorphisms of the circle and hyperbolic curvature, Conform. Geom. Dyn. 5 (2001)1-5.

    [8] B. Ou, An equality for curvature function of a simple closed curve on the plane, International J. of Math. and Math. Sci 49(1) (2003), 3115-3122.

    [9] M. P. do Carmo, Remannian geometry, Birkha user Boston 1992.

    [10] J. Jost, Nonpositive curvature: geometric and analytic aspects, Lectures in Math., Birkh user Boston 1997.

    [11] R.P. Boas, Jr., A differential inequality, Bull. A.M.S. 51 (1945), 95-96

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