常曲率曲面中凸集的等周亏格的上界估计
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摘要
本文首先给出李群成为幺模群的判定定理以及齐性空间上存在运动密度的判定定理的新证明.接下来本文证明了常曲率空间Xcn的等距变换群I(Xcn)正好是以正交群O(n；R)为结构群的主丛O(Xcn).通过计算,证明了I(Xcn)是幺模群.这个结果也给出了活动标架法的一个解释.进一步本文研究了常曲率空间中具有两个常主曲率的齐性超曲面构成的齐性空间,给出了该齐性空间上的运动密度.
令K是常曲率曲面Xc2上的区域,A和L分别记K的面积和(?)K的周长.关于K的等周不等式为L2-4πA+cA2≥0.等号成立当且仅当K为测地圆盘.数量△(K)=L2-4πA+cA2称为K的等周亏格.常曲率曲面上等周亏格的下界已由D.Klain,J.Zhou和F.Chen得到.而等周亏格的上界估计的相关结果较少,对于平面上的卵形区域,O.Bottema给出了等周亏格的一个上界.本文将该结果推广到了常曲率曲面的情形,具体地我们得到了下面的定理：
定理.设K是Xc2中的凸体,具有光滑的严格凸的边界(?)K.当c<0时,假设(?)K的曲率对于任意的点x∈(?)K,设x处的曲率半径为ρ(x).ρ的最大值和最小值分别记为ρM和ρm,则有等号成立当且仅当K是测地圆盘.另外我们也有下面的不等式
最后本文给出了Blaschke滚动定理在常曲率曲面情形的一个新证明.该定理对于等周亏格的上界估计是必要的.
This dissertation investigates firstly the sufficient and necessary conditions for Lie groups to be of unimodula, and for homogeneous spaces to have invariant measure. We give new proofs of these properties. Then we prove that the group I(Xcn) of isometries of space form Xcn is unimodular and indeed the principle bundle O(Xcn) with structure group O(n; R). It also gives an explanation of the moving frame methods in Xcn. Furthermore, we derive the kinematic density of a homogeneous space of a class of homogeneous hypersurface in Xcn with two constant principal curvatures.
Let K be a domain in the constant curvature surface Xc2, A and L be the area of K and the length of (?)K, respectively. Then the isoperimetric inequality of K is L2-4πA+cA2≥0. The equality sign holds if and only if K is a geodesic disc. The quantityΔ(K)=L2-4πA+cA2 is called the isoperimetric deficit of K. The lower bound of the isoperimetric deficit have been obtained by D. Klain, J. Zhou and F. Chen. We know little about the upper bound of the isoperimetric deficit. In the plane case,O. Bottema given an upper bound of the isoperimetric deficit for an oval domain. We generalize the result to the case Xc2.
Theorem. Let K be a convex domain in Xc2 with smooth and strictly convex boundary (?)K. When c<0, we assume that the geodesic curvature of (?)K satisfyρ(x) denotes the curvature radius at x∈(?)K. LetρM andρm be the maximum and minimum of p respectively, then we have The equality sign holds if and only if K is a geodesic disc. We also have
At the end, we give a new proof of the Blaschke's rolling theorem in Xc2. The theorem is necessary for the estimating of the upper bound of the isoperimetric deficit.
令K是常曲率曲面Xc2上的区域,A和L分别记K的面积和(?)K的周长.关于K的等周不等式为L2-4πA+cA2≥0.等号成立当且仅当K为测地圆盘.数量△(K)=L2-4πA+cA2称为K的等周亏格.常曲率曲面上等周亏格的下界已由D.Klain,J.Zhou和F.Chen得到.而等周亏格的上界估计的相关结果较少,对于平面上的卵形区域,O.Bottema给出了等周亏格的一个上界.本文将该结果推广到了常曲率曲面的情形,具体地我们得到了下面的定理：
定理.设K是Xc2中的凸体,具有光滑的严格凸的边界(?)K.当c<0时,假设(?)K的曲率对于任意的点x∈(?)K,设x处的曲率半径为ρ(x).ρ的最大值和最小值分别记为ρM和ρm,则有等号成立当且仅当K是测地圆盘.另外我们也有下面的不等式
最后本文给出了Blaschke滚动定理在常曲率曲面情形的一个新证明.该定理对于等周亏格的上界估计是必要的.
This dissertation investigates firstly the sufficient and necessary conditions for Lie groups to be of unimodula, and for homogeneous spaces to have invariant measure. We give new proofs of these properties. Then we prove that the group I(Xcn) of isometries of space form Xcn is unimodular and indeed the principle bundle O(Xcn) with structure group O(n; R). It also gives an explanation of the moving frame methods in Xcn. Furthermore, we derive the kinematic density of a homogeneous space of a class of homogeneous hypersurface in Xcn with two constant principal curvatures.
Let K be a domain in the constant curvature surface Xc2, A and L be the area of K and the length of (?)K, respectively. Then the isoperimetric inequality of K is L2-4πA+cA2≥0. The equality sign holds if and only if K is a geodesic disc. The quantityΔ(K)=L2-4πA+cA2 is called the isoperimetric deficit of K. The lower bound of the isoperimetric deficit have been obtained by D. Klain, J. Zhou and F. Chen. We know little about the upper bound of the isoperimetric deficit. In the plane case,O. Bottema given an upper bound of the isoperimetric deficit for an oval domain. We generalize the result to the case Xc2.
Theorem. Let K be a convex domain in Xc2 with smooth and strictly convex boundary (?)K. When c<0, we assume that the geodesic curvature of (?)K satisfyρ(x) denotes the curvature radius at x∈(?)K. LetρM andρm be the maximum and minimum of p respectively, then we have The equality sign holds if and only if K is a geodesic disc. We also have
At the end, we give a new proof of the Blaschke's rolling theorem in Xc2. The theorem is necessary for the estimating of the upper bound of the isoperimetric deficit.
引文
[1]Santalo L. A., Integral Geometry and Geometric Probability, U.S.A.: Addison-Wesley, Advanced Book Program Reading, Massachusetts,1976.
[2]Ren D., Topics in Integral Geometry, Singapore:World Scientific,1994.
[3]Weil A., L'integration dans les groupes totologiques et ses applications, Actualites Sci. Indust.1938,869.
[4]Weil A., Review of Chern's article, Math. Reviews,1942,3:253.
[5]Chern S. S., On integral geometry in Klein spaces, Ann. of Math.,1942, 43(2):178-189.
[6]Chern S. S., On the kinematic formula in integral geometry, J. Math. and Mech.1966,16(1):101-118
[7]Howard R., The kinematic formula in Riemannian homogeneous spaces, Mem. AMS.,1993,509:1-69.
[8]Zhou J., Kinematic formulas for mean curvature powers of hyp ersurfaces and Hadwiger's theorem in R2n, Trans. of the AMS.1994,345(1):243-262.
[9]Schneider R., Convex Bodies:The Brunn-Minkowski Theory, Cambridge University Press,1993.
[10]Zhang G., A sufficient condition for one convex body containing another, Chinese Ann. of Math.(Series B) 1988,4:447-451.
[11]Zhou J., Sufficient conditions for one domain to contain another in a space of constant curvature, Proc. of the AMS.1998,126:2797-2803.
[12]Zhou J., The sufficient condition for a convex body to enclose another in R4, Proc. of the AMS.1994,121(3):907-913.
[13]Zhou J., When can one domain enclose another in R3, journal of the Aus-tralian Mathematical society (Series A),1995,59:266-272.
[14]Burago Y. D., Zalgaller V. A., Geometric Inequalities, Germany:Springer-Verlag,1988.
[15]Grinberg E., Ren D., Zhou J., The symmetric isoperimetric deficit and the containment problem in a plane of constant curvature, preprint.
[16]Hsiung W. Y., An elementary proof of the isoperimetric problem, Chinese Ann. Math.,2002,23A(1):7-12.
[17]Osserman R., The isoperimetric inequality, Bull. Amer. Math. Soc.,1978, 84(6):1182-1238.
[18]Osserman R., Bonnesen-style isoperimetric inequality, Amer. Math. Monthly,1979,86(1):1-29.
[19]Zhou J., On Bonnesen-type inequalities, Acta Math. Sinica,2007,50(6): 1397-1402.
[20]Zhou J., On the Willmore deficit of convex surface, Lectures in Appl. Math. of Amer. Math. Soc.,1994,8:279-287.
[21]Zhou J., Xia Y., Zeng C. Some new Bonnesen-style inequalities, to appear in J. Korean Math. Soc..
[22]Zhou J., Chen F., The Bonnesen-type inequality in a plane of constant cur-vature, Journal of Korean Math. Soc.,2007,44(6):1363-1372.
[23]Klain D., Bonnesen-type inequalities for surfaces of constant curvature, Ad-vances in Applied Mathematics,2007,39(2):143-154.
[24]Zhang G., Zhou J., Containment measures in integral geometry, Integral Geometry and Convexity, Singapore:World Scientific,2005:153-168.
[25]Zhang G., Geometric inequalities and inclusion measures of convex bodies, Mathematika,1994,41:95-116.
[26]Zhou J., Zhou C., Ma F., Isoperimetric deficit upper limit of a planar con-vex set, rendiconti del Circolo matemetico di Palermo Serie Ⅱ,2009, Suppl. 81:363-367.
[27]Bottema O., Eine obere Grenze fur das isoperimetrische Defizit ebener Kur-ven, Nederl. Akad. Wetensch. Proc.,1933, A66:442-446.
[28]Blaschke W., Kreis und Kugel, Auflage, Berlin,1956.
[29]Koutroufiotis D., On Blaschke's rolling theorem, Arch. Math.,1972,23:655-660.
[30]Rauch J., An inclusion theorem for ovaloids with comparable second funda-mental forms, J. Diff. Geom.,1974,9:501-505.
[31]Delgado J., Blaschke's theorem for convex hypersurfaces, J. Diff. Geom., 1979,14:489-496.
[32]Howard R., Blaschke's rolling theorem for manifolds with boundary, manuscripta math.,1999,99:471-483.
[33]陈省身,陈维桓,微分几何讲义(第二版),北京：北京大学出版社,2001.
[34]Helgason S., Group and geometric analysis, U.S.A.:Academic Press, INC., 1984.
[35]Helgason S., Differential geometry, Lie groups, and symmetric spaces, U.S.A. Academic Press, INC.,1978.
[36]Palais R. S., Terng C., Critical point theory and submanifold geometry, Lecture Notes in Mathematics 1353,1988.
[37]Berndt J., Console S., Olmos C., Submanifolds and holonomy, U.S.A. CHAPMAN&HALL/CRC,2003.
[38]Kobayashi S., Transformation groups in differential geometry, Germany: Springer-Verlag,1995.
[39]Escudero C. A., Reventos A., Solanes G., Focal sets in two-dimensional space forms, Pacific J. Math.,2007,233(2):309-320.
[40]do Carmo M. P., Differentiable Curves and Surfaces, New Jersey:Prentice-Hall,1976.
[41]张筑生,微分拓扑新讲,北京：北京大学出版社,2002.
[42]Hirsch M. W., Differential Topology, New York:Springer-Verlag,1976.
[43]Fulton W., Algebraic Topology, A First Course, New York:Springer-Verlag, 1995.
[44]do Carmo M. P., Riemannian Geometry, Boston:Birkhauser,1992.
[45]Vidal Abascal E., A generalization of Steiner's formulae, Bull. Amer. Math. Soc.,1947,53(8):841-844.
[46]Eisenhart L., An Introduction to Differential Geometry, Princeton, N. J.:Princeton University Press,1940.
[47]Allendoerfer C. B., Steiner's formulae on a general Sn+1, Bull. Amer. Math. Soc.,1948,54(2):128-135.
[2]Ren D., Topics in Integral Geometry, Singapore:World Scientific,1994.
[3]Weil A., L'integration dans les groupes totologiques et ses applications, Actualites Sci. Indust.1938,869.
[4]Weil A., Review of Chern's article, Math. Reviews,1942,3:253.
[5]Chern S. S., On integral geometry in Klein spaces, Ann. of Math.,1942, 43(2):178-189.
[6]Chern S. S., On the kinematic formula in integral geometry, J. Math. and Mech.1966,16(1):101-118
[7]Howard R., The kinematic formula in Riemannian homogeneous spaces, Mem. AMS.,1993,509:1-69.
[8]Zhou J., Kinematic formulas for mean curvature powers of hyp ersurfaces and Hadwiger's theorem in R2n, Trans. of the AMS.1994,345(1):243-262.
[9]Schneider R., Convex Bodies:The Brunn-Minkowski Theory, Cambridge University Press,1993.
[10]Zhang G., A sufficient condition for one convex body containing another, Chinese Ann. of Math.(Series B) 1988,4:447-451.
[11]Zhou J., Sufficient conditions for one domain to contain another in a space of constant curvature, Proc. of the AMS.1998,126:2797-2803.
[12]Zhou J., The sufficient condition for a convex body to enclose another in R4, Proc. of the AMS.1994,121(3):907-913.
[13]Zhou J., When can one domain enclose another in R3, journal of the Aus-tralian Mathematical society (Series A),1995,59:266-272.
[14]Burago Y. D., Zalgaller V. A., Geometric Inequalities, Germany:Springer-Verlag,1988.
[15]Grinberg E., Ren D., Zhou J., The symmetric isoperimetric deficit and the containment problem in a plane of constant curvature, preprint.
[16]Hsiung W. Y., An elementary proof of the isoperimetric problem, Chinese Ann. Math.,2002,23A(1):7-12.
[17]Osserman R., The isoperimetric inequality, Bull. Amer. Math. Soc.,1978, 84(6):1182-1238.
[18]Osserman R., Bonnesen-style isoperimetric inequality, Amer. Math. Monthly,1979,86(1):1-29.
[19]Zhou J., On Bonnesen-type inequalities, Acta Math. Sinica,2007,50(6): 1397-1402.
[20]Zhou J., On the Willmore deficit of convex surface, Lectures in Appl. Math. of Amer. Math. Soc.,1994,8:279-287.
[21]Zhou J., Xia Y., Zeng C. Some new Bonnesen-style inequalities, to appear in J. Korean Math. Soc..
[22]Zhou J., Chen F., The Bonnesen-type inequality in a plane of constant cur-vature, Journal of Korean Math. Soc.,2007,44(6):1363-1372.
[23]Klain D., Bonnesen-type inequalities for surfaces of constant curvature, Ad-vances in Applied Mathematics,2007,39(2):143-154.
[24]Zhang G., Zhou J., Containment measures in integral geometry, Integral Geometry and Convexity, Singapore:World Scientific,2005:153-168.
[25]Zhang G., Geometric inequalities and inclusion measures of convex bodies, Mathematika,1994,41:95-116.
[26]Zhou J., Zhou C., Ma F., Isoperimetric deficit upper limit of a planar con-vex set, rendiconti del Circolo matemetico di Palermo Serie Ⅱ,2009, Suppl. 81:363-367.
[27]Bottema O., Eine obere Grenze fur das isoperimetrische Defizit ebener Kur-ven, Nederl. Akad. Wetensch. Proc.,1933, A66:442-446.
[28]Blaschke W., Kreis und Kugel, Auflage, Berlin,1956.
[29]Koutroufiotis D., On Blaschke's rolling theorem, Arch. Math.,1972,23:655-660.
[30]Rauch J., An inclusion theorem for ovaloids with comparable second funda-mental forms, J. Diff. Geom.,1974,9:501-505.
[31]Delgado J., Blaschke's theorem for convex hypersurfaces, J. Diff. Geom., 1979,14:489-496.
[32]Howard R., Blaschke's rolling theorem for manifolds with boundary, manuscripta math.,1999,99:471-483.
[33]陈省身,陈维桓,微分几何讲义(第二版),北京：北京大学出版社,2001.
[34]Helgason S., Group and geometric analysis, U.S.A.:Academic Press, INC., 1984.
[35]Helgason S., Differential geometry, Lie groups, and symmetric spaces, U.S.A. Academic Press, INC.,1978.
[36]Palais R. S., Terng C., Critical point theory and submanifold geometry, Lecture Notes in Mathematics 1353,1988.
[37]Berndt J., Console S., Olmos C., Submanifolds and holonomy, U.S.A. CHAPMAN&HALL/CRC,2003.
[38]Kobayashi S., Transformation groups in differential geometry, Germany: Springer-Verlag,1995.
[39]Escudero C. A., Reventos A., Solanes G., Focal sets in two-dimensional space forms, Pacific J. Math.,2007,233(2):309-320.
[40]do Carmo M. P., Differentiable Curves and Surfaces, New Jersey:Prentice-Hall,1976.
[41]张筑生,微分拓扑新讲,北京：北京大学出版社,2002.
[42]Hirsch M. W., Differential Topology, New York:Springer-Verlag,1976.
[43]Fulton W., Algebraic Topology, A First Course, New York:Springer-Verlag, 1995.
[44]do Carmo M. P., Riemannian Geometry, Boston:Birkhauser,1992.
[45]Vidal Abascal E., A generalization of Steiner's formulae, Bull. Amer. Math. Soc.,1947,53(8):841-844.
[46]Eisenhart L., An Introduction to Differential Geometry, Princeton, N. J.:Princeton University Press,1940.
[47]Allendoerfer C. B., Steiner's formulae on a general Sn+1, Bull. Amer. Math. Soc.,1948,54(2):128-135.
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