R～n中曲线论初步
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摘要
本文按照R~3经典微分几何的研究思路，对R~n中的正则曲线作了初步探讨，我们首先从R~n中正则曲线的定义入手，然后介绍了正则曲线的自然参数表示和Frenet-Serret公式，接下来利用这一重要公式给出了R~n中正则曲线的各阶曲率的计算公式，讨论了R~n中的一般螺旋线和曲线的平坦性，我们还对R~n中的闭曲线证明了一个几何不等式。
In this thesis,following the ideas of classical differential geometry in R~3,we discuss regular curves in R~n.From the beginning of definition of regular curves,we introduce the natural para-metrization and Frenet-Serret Formula of regular curves.Then,by the Formula,we give calculation formulas of regular curves' curvatures ,and discuss the general helices and flatness of curves in R~n,finally we prove a geometry inequality of curvatures and arc length.
In this thesis,following the ideas of classical differential geometry in R~3,we discuss regular curves in R~n.From the beginning of definition of regular curves,we introduce the natural para-metrization and Frenet-Serret Formula of regular curves.Then,by the Formula,we give calculation formulas of regular curves' curvatures ,and discuss the general helices and flatness of curves in R~n,finally we prove a geometry inequality of curvatures and arc length.
引文
[1] Yu. Aminov, Differential Geometry and Topology of Curves, Amsterdam, the Netherlands: Gordon and Breach Science Publishers, 2000.
[2] M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, N. J., 1976. (有中译本)
[3] F. J. Craveiro De Carvalho, A Theorem of F. R. Fabricius-Bjerre on Helices, Pre-Publicacoes do Departamento de Matematica Universidade de Coimbra, Preprint Number 03-01 .
[4] M. Ghomi, Classical open problems in differential geometry, last revised October 21, 2004. (http://www.math.gatech.edu/ghomi/Papers/op.pdf)
[5] V. A. Gorkaviy, One integral inequality for closed curves in Euclidean space, CRAS Paris, 321(1995), 1587-1591.
[6] C. C. Hsiung, A First Course in Differential Geometry, Pure & Applied Math., Wiley, New York, 1981.
[7] W. Klingenberg, A Course in Differential Geometry, GTM 51, Springer, Berlin, 2nd edition, 1983.
[8] Y. S. Kupitz & M. A. Perles, A condition for flatness of curves in (？)~n, Amer. Math. Monthly, 97(1990), 401-405.
[9] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer-Verlag, New York,Inc, 1997.
[10] S. Lipschutz, Theory and Problems of Differential Geometry, Schaum's Outline Series, McGraw-Hill, 1969.
[11] R. S. Millman & G. D. Parker, Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977.
[12] J. Oprea, Differential Geometry and Its Applications, Prentice Hall, 1997.
[13] S. L. Pan, Lectures on Differential Geometry in R~3, Department of Mathematics, East China Normal University, 2005.
[14] A. Pressley, Elementary Differential Geometry, Springer, 2001.
[15] J. L. Weiner, An inequality involving the length, curvation, and torsion torsions of a curve in Euclidean n-space, Pacific J. of Math, 74(1978).
[16] J. L. Weiner, How helical can a closed, twisted space curve be? Amer. Math. Monthly, 107(2000), 327-333.
[17] 苏步青、胡和生、沈纯理、潘养廉、张国樑著，微分几何，人民教育出版社，1980年1月．
[18] 吴大任，微分几何讲义，(第四版)，人民教育出版社，1981年10月．
[2] M. P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, Englewood Cliffs, N. J., 1976. (有中译本)
[3] F. J. Craveiro De Carvalho, A Theorem of F. R. Fabricius-Bjerre on Helices, Pre-Publicacoes do Departamento de Matematica Universidade de Coimbra, Preprint Number 03-01 .
[4] M. Ghomi, Classical open problems in differential geometry, last revised October 21, 2004. (http://www.math.gatech.edu/ghomi/Papers/op.pdf)
[5] V. A. Gorkaviy, One integral inequality for closed curves in Euclidean space, CRAS Paris, 321(1995), 1587-1591.
[6] C. C. Hsiung, A First Course in Differential Geometry, Pure & Applied Math., Wiley, New York, 1981.
[7] W. Klingenberg, A Course in Differential Geometry, GTM 51, Springer, Berlin, 2nd edition, 1983.
[8] Y. S. Kupitz & M. A. Perles, A condition for flatness of curves in (？)~n, Amer. Math. Monthly, 97(1990), 401-405.
[9] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, Springer-Verlag, New York,Inc, 1997.
[10] S. Lipschutz, Theory and Problems of Differential Geometry, Schaum's Outline Series, McGraw-Hill, 1969.
[11] R. S. Millman & G. D. Parker, Elements of Differential Geometry, Prentice-Hall Inc., Englewood Cliffs, New Jersey, 1977.
[12] J. Oprea, Differential Geometry and Its Applications, Prentice Hall, 1997.
[13] S. L. Pan, Lectures on Differential Geometry in R~3, Department of Mathematics, East China Normal University, 2005.
[14] A. Pressley, Elementary Differential Geometry, Springer, 2001.
[15] J. L. Weiner, An inequality involving the length, curvation, and torsion torsions of a curve in Euclidean n-space, Pacific J. of Math, 74(1978).
[16] J. L. Weiner, How helical can a closed, twisted space curve be? Amer. Math. Monthly, 107(2000), 327-333.
[17] 苏步青、胡和生、沈纯理、潘养廉、张国樑著，微分几何，人民教育出版社，1980年1月．
[18] 吴大任，微分几何讲义，(第四版)，人民教育出版社，1981年10月．