球面子流形的光锥对偶
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摘要
本文在Minkowski空间中伪球的Legendrian对偶理论框架下,研究了光球中的子流形.本文研究的内容包括2维光球中的曲线,3维光球中的曲线以及n维光球中的超曲面等问题.光球可以嵌入到光锥和de Sitter空间中.然后我们分别考虑了光锥和光锥之间的对偶以及光锥和de Sitter空间的对偶,从而得到了光球中子流形的光锥对偶超曲面和相应欧氏球面中的的子流形的光锥对偶超曲面两类超曲面.我们研究了它们的奇点并进行了比较,得到了一些有趣的结果.我们还给出了这些曲面的奇点分类.
本文结构如下:
在第一章引言部分,我们介绍了关于光球中的子流形背景知识和发展概况.其次,我们对全文的安排及主要研究内容做了说明.
在第二章中,我们给出了本文研究所需的预备知识.
在第三章中,我们主要介绍了2维光球中的曲线,我们给出了2维光球中曲线的Frenet-Serret型公式,在此公式的基础上我们给出该曲线的光锥对偶超曲面和对应的单位欧氏球面上的曲线的光锥对偶超曲面.我们主要给出了这两类曲面的奇点分类.
在第四章中,我们主要研究了3维光球中的曲线,我们计算出3维光球中曲线的Frenet-Serret型公式,在此公式的基础上我们给出该曲线的光锥对偶超曲面和对应的欧氏单位球面上的曲线的光锥对偶超曲面并给出它们的奇点分类.
在第五章中,我们研究了n维光球中的超曲面.我们给出了n维光球中的超曲面的光锥对偶超曲面和对应的n维欧氏单位球面中的超曲面的光锥对偶超曲面.我们计算出了它们的临界点集并将其定义为两类超曲面的光锥焦曲面.作为Legendrian奇点理论的应用,我们还研究了球面中的超曲面和抛物(n1)维球面以及抛物n维超二次型的切触状况.
In this thesis, we studied the submanifolds in the lightcone unit sphere in the frame-work of the theory of Legendrian dualities between pseudo-spheres in Minkowski space.The subjects we studied include the curves in the unit2-sphere, the curves in the unit3-sphere and the hypersurfaces in the unit n-sphere. The lightcone unit sphere can benaturally embedded in the lightcone and the de Sitter space. We consider the Legendriandualities between lightcone and lightcone and the Legendrian dualities between lightconeand de Sitter space. Then we got the lightcone dual hypersurface of the submanifold inthe lightcone unit sphere and the lightcone dual hypersurface of the submanifold in theEuclidean unit sphere. We studied the singularities of these hypersurfaces. When we com-pared these singularities, we got some interesting results. We also gave the classifcationsof the singularities of these hypersurfaces.
The thesis is organized as follows.
In Chapter1, we frst reviewed briefy the background of the subject of lightconeunit sphere and introduced the outline of development of this subject in recent years.Moreover, we introduced the structure of the full thesis and described the main contentof this thesis.
In Chapter2, we introduced the basic preliminary knowledge in this thesis.
In Chapter3, we mainly introduced the curves in the unit2-sphere. We gave theFrenet-Serret type formulae for the curve in the unit2-sphere. According to the formulae,we gave the lightcone dual hypersurface of the curve in the lightcone unit2-sphere andthe lightcone dual hypersurfaces of the curve in the Euclidean unit2-sphere. We gave theclassifcations of the singularities of these two lightcone dual hypersurfaces.
In Chapter4, we mainly studied the curves in the unit3-sphere. We gave the Frenet-Serret type formulae for the curve in the unit3-sphere. According to the formulae, wegave the lightcone dual hypersurfaces of the curve in the lightcone unit3-sphere and the lightcone dual hypersurfaces of the curve in the Euclidean unit3-sphere. As the mainresult, we gave the classifcations of the singularities of these lightcone dual hypersurfaces.
In Chapter5, we mainly studied the hypersurfaces in the unit n-sphere. We gavethe lightcone dual hypersurface of the hypersurface in the lightcone unit n-sphere andthe lightcone dual hypersurface of the hypersurface in the Euclidean unit n-sphere. Wecalculated the critical set of these hypersurfaces and we defned them as the lightcone focalsurface of the hypersurfaces respectively. We also studied the contact of hypersurfaceswith the parabolic (n1)-spheres and parabolic n-hyperquadric as an application of thetheory of Legendrian singularities.
本文结构如下:
在第一章引言部分,我们介绍了关于光球中的子流形背景知识和发展概况.其次,我们对全文的安排及主要研究内容做了说明.
在第二章中,我们给出了本文研究所需的预备知识.
在第三章中,我们主要介绍了2维光球中的曲线,我们给出了2维光球中曲线的Frenet-Serret型公式,在此公式的基础上我们给出该曲线的光锥对偶超曲面和对应的单位欧氏球面上的曲线的光锥对偶超曲面.我们主要给出了这两类曲面的奇点分类.
在第四章中,我们主要研究了3维光球中的曲线,我们计算出3维光球中曲线的Frenet-Serret型公式,在此公式的基础上我们给出该曲线的光锥对偶超曲面和对应的欧氏单位球面上的曲线的光锥对偶超曲面并给出它们的奇点分类.
在第五章中,我们研究了n维光球中的超曲面.我们给出了n维光球中的超曲面的光锥对偶超曲面和对应的n维欧氏单位球面中的超曲面的光锥对偶超曲面.我们计算出了它们的临界点集并将其定义为两类超曲面的光锥焦曲面.作为Legendrian奇点理论的应用,我们还研究了球面中的超曲面和抛物(n1)维球面以及抛物n维超二次型的切触状况.
In this thesis, we studied the submanifolds in the lightcone unit sphere in the frame-work of the theory of Legendrian dualities between pseudo-spheres in Minkowski space.The subjects we studied include the curves in the unit2-sphere, the curves in the unit3-sphere and the hypersurfaces in the unit n-sphere. The lightcone unit sphere can benaturally embedded in the lightcone and the de Sitter space. We consider the Legendriandualities between lightcone and lightcone and the Legendrian dualities between lightconeand de Sitter space. Then we got the lightcone dual hypersurface of the submanifold inthe lightcone unit sphere and the lightcone dual hypersurface of the submanifold in theEuclidean unit sphere. We studied the singularities of these hypersurfaces. When we com-pared these singularities, we got some interesting results. We also gave the classifcationsof the singularities of these hypersurfaces.
The thesis is organized as follows.
In Chapter1, we frst reviewed briefy the background of the subject of lightconeunit sphere and introduced the outline of development of this subject in recent years.Moreover, we introduced the structure of the full thesis and described the main contentof this thesis.
In Chapter2, we introduced the basic preliminary knowledge in this thesis.
In Chapter3, we mainly introduced the curves in the unit2-sphere. We gave theFrenet-Serret type formulae for the curve in the unit2-sphere. According to the formulae,we gave the lightcone dual hypersurface of the curve in the lightcone unit2-sphere andthe lightcone dual hypersurfaces of the curve in the Euclidean unit2-sphere. We gave theclassifcations of the singularities of these two lightcone dual hypersurfaces.
In Chapter4, we mainly studied the curves in the unit3-sphere. We gave the Frenet-Serret type formulae for the curve in the unit3-sphere. According to the formulae, wegave the lightcone dual hypersurfaces of the curve in the lightcone unit3-sphere and the lightcone dual hypersurfaces of the curve in the Euclidean unit3-sphere. As the mainresult, we gave the classifcations of the singularities of these lightcone dual hypersurfaces.
In Chapter5, we mainly studied the hypersurfaces in the unit n-sphere. We gavethe lightcone dual hypersurface of the hypersurface in the lightcone unit n-sphere andthe lightcone dual hypersurface of the hypersurface in the Euclidean unit n-sphere. Wecalculated the critical set of these hypersurfaces and we defned them as the lightcone focalsurface of the hypersurfaces respectively. We also studied the contact of hypersurfaceswith the parabolic (n1)-spheres and parabolic n-hyperquadric as an application of thetheory of Legendrian singularities.
引文
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[12] Gao S P, Li Y C., The unfolding of equivariant bifurcation problems with parameterssymmetry[J]. Acta Math Sci Ser B,2004,24(4):623–632.
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[2] Chen L., On spacelike surfaces in Anti de Sitter3-space from the contact view-point[J]. Hokkaido Math. J. Volume38, Number4(2009),701-720.
[3] C. McCrory and T. Shifrin, Cusps of the projective Gauss map, J. Diferential Geom.19(1984), no.1,257-276.
[4] C. McCrory, T. Shifrin, and R. Varley, The Gauss map of a generic hypersurface inP4, J. Diferential Geom.30(1989), no.3,689-759.
[5] Che M G, Jiang Y, Pei D H., The hyperbolic Darboux image and rectifying Gaussiansurface of nonlightlike curve in Minkowski3-space[J]. J Math Res Exposition,2008,28(3):651–658.
[6] Davydov A A, Ishikawa G, Izumiya S, et al., Generic singularities of implicit systemsof frst order diferential equations on the plane[J]. Jpn J Math2008,3(1):93–119.
[7] D. Bleecker and L. Wilson, Stability of Gauss maps, Illinois J. Math.22(1978), no.2,279-289.
[8] E. E. Landis, Tangential singularities, Funktsional. Anal. i Prilozhen.15(1981), no.2,36-49,96(Russian). MR617468. English translation: Functional Anal. Appl.15(1981), no.2,103-114.
[9] F. Finster, Light-cone expansion of the Dirac sea in the presence of chiral and scalarpotentials, J. Math. Phys.,41,6689(2000).
[10] F. Schwarz, Asymptotic Expansions of Fourier Integrals with Light Cone Singular-ities, J. Math. Phys.,13,1621(1972).
[11] Fusho T, Izumiya S., Lightlike surfaces of spacelike curves in de Sitter3-space[J]. JGeom,2008,88(1-2),19–29.
[12] Gao S P, Li Y C., The unfolding of equivariant bifurcation problems with parameterssymmetry[J]. Acta Math Sci Ser B,2004,24(4):623–632.
[13] Gao S P, Li Y C., Unfolding of multiparameter equivariant bifurcation problemswith respect to left-right equivalence[J]. Chinese Ann Math Ser A(Chinese),2003,24(3):341–348.
[14] H. L. Liu, S. D. Jung, Hypersurfaces in lightlike cone, J. Geom. Phys.58(2008),913–922.
[15] I. R. Porteous, Some remrks on duality in S3, Geometry and topology of caustics,Banach Center Publ.50, Polish Acad. Sci., Warsaw,(2004),217–226.
[16] Izumiya S, Diferential Geometry from the viewpoint of Lagrangian or Legendriansingularity theory, in Singularity Theory (ed., D. Ch′eniot et al), World Scientifc(2007),241–275.
[17] Izumiya S, Lengendrian dualities and spacelike hypersurfaces in the lightcone, Mosc.Math. J.9(2009)325–357.
[18] Izumiya S, Jiang Y and Pei D H, Lightcone dualities for curves in the sphere, Quart.J. Math.64(2013),221–234.
[19] Izumiya S, Jiang Y and Pei D H, Lightcone dualities for hypersurface in the sphere,preprint (2013).
[20] Izumiya S, Jiang Y and Sato T, Lightcone dualities for curves in the3-sphere, toappear in Journal of Mathematical and Physics (2012).
[21] Izumiya S, Kikuchi M, Takahashi M., Global properties of spacelike curves inMinkowski3-space[J]. J Knot Theory Ramifcations,2006,15(7):869–881.
[22] Izumiya S, Kossowski M, Pei D H, et al., Singularities of lightlike hypersurfaces inMinkowski four-space[J]. Tohoku Math J,2006,58(1):71–88.
[23] Izumiya S, Pei D H, Romero-Fuster M C. Takahashi M., On the horospherical ridgesof submanifolds of codimension2in hyperbolic n-space[J]. Bull Braz Math Soc,2004,35(2):177–198.
[24] Izumiya S, Pei D H, Romero-Fuster M C, et al., The horospherical geometry ofsubmanifolds in hyperbolic space[J]. J London Math Soc,2005,71(3):779–800.
[25] Izumiya S, Pei D H, Romero-Fuster M C., The horospherical geometry of surfacesin hyperbolic4-space[J]. Israel J Math,2006,154:361–379.
[26] Izumiya S, Pei D H, Romero-Fuster M C., The lightcone Gauss map of a spacelikesurface in Minkowski4-space[J]. Asian J Math,2004,8(3):511–530.
[27] Izumiya S, Pei D H, Romero-Fuster M C., Umbilicity of space-like submanifolds ofMinkowski space[J]. Proc Roy Soc Edinburgh Sect A,2004,134(2):375-387.
[28] Izumiya S, Pei D H, Sano T., Horospherical surfaces of curves in Hyperbolic space[J].Publ Math Debrecen,2004,64(1-2):1-13.
[29] Izumiya S, Pei D H, Sano T., The lightcone Gauss map and the lightcone developableof a spacelike curve in Minkowski3-space[J]. Glasgow Math J,2000,42(1):75–89.
[30] Izumiya S, Pei D H, Takahashi M., Singularities of evolutes of hypersurfaces inhyperbolic space[J]. Proc Edinb Math Soc,2004,47(1):131–153.
[31] Izumiya S, Pei D H, and Sano T, Singularities of hyperbolic Gauss maps, Proc.London Math. Soc.(3)86(2003), no.2,485-512.
[32] Izumiya S, Romero-Fuster M C., The horospherical Gauss-Bonnet type theorem inhyperbolic space[J]. J Math Soc Japan,2006,58(4):965–984.
[33] Izumiya S, Romero-Fuster M C., The lightlike fat geometry on spacelike submani-folds of codimension two in Minkowski space[J]. Selecta Math,2007,13(1):23–55.
[34] Izumiya S, Saji K, Takeuchi N., Circular surfaces[J]. Adv Geom,2007,7(2):295–313.
[35] Izumiya S, Takahashi M., Spacelike parallels and evolutes in Minkowski pseudo-spheres[J]. J Geom Phys,2007,57(8):1569–1600.
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