度量的变分及其在黎曼几何中的应用
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摘要
变分法是17世纪末发展起来的一门数学分支。其理论完整,在力学、物理学、光学、摩擦学、经济学、宇航理论、信息论和自动控制论等诸多方面有广泛的应用。我们可以看到变分法在经典微分几何中的重要作用。有些文献通过活动标架得到关于度量的黎曼曲率张量、里奇曲率张量以及数量曲率张量的第一、第二变分公式,从而得到度量的体积变分与子流形的体积变分之间的关系。有了这些公式,我们把这些公式应用于作用在1-形式上热不变量的变分。从而得出一个结论：对于4维紧致黎曼流形M,如果度量g是共形平坦并且是热核不变量的临界点,那么M是数量平坦或空间形式。
The variational method is a branch of mathematics, and was developed in the 17th century. Its theory is complete and has extensive application in mechanics, physics, optics, tribological, economics, aerospace theory and automatic control theory, etc. We can also see the variational method plays an important role in classical differential geometry. Some literatures give the first and the second variation formulas of the Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using the moving frame method, so there is a relation between the variation of the volume of a metric and that of a submanifold. We give an application of these formulas to the variations of heat invariants which function on one-form. We prove that a conformally flat metric g is a critical point of the third heat invariant functional for a compact 4-dimensional manifold M, then (M, g) is either scalar flat or a space form.
The variational method is a branch of mathematics, and was developed in the 17th century. Its theory is complete and has extensive application in mechanics, physics, optics, tribological, economics, aerospace theory and automatic control theory, etc. We can also see the variational method plays an important role in classical differential geometry. Some literatures give the first and the second variation formulas of the Riemannian curvature tensor, Ricci curvature tensor and scalar curvature of a metric by using the moving frame method, so there is a relation between the variation of the volume of a metric and that of a submanifold. We give an application of these formulas to the variations of heat invariants which function on one-form. We prove that a conformally flat metric g is a critical point of the third heat invariant functional for a compact 4-dimensional manifold M, then (M, g) is either scalar flat or a space form.
引文
[1]牛庠均：现代变分原理,北京：北京工业大学出版社,1992,22-55
[2]Hu,Z.J.,Li H.Z. A new variational characterization of n-dimensional space forms. Trans. Amer. Math.Soc.,356(8),2004,3005-3023
[3]Fa En WU.On the Variation of a metric and its Application. The Editorial Office of AMS and Springer-Verlag,2009
[4]老大中：变分法基础(第2版),北京：国防工业出版社,2006,45-96
[5]陈维桓,李兴校：黎曼儿何引论上册,北京：北京大学出版社2002,171-257
[6]彭家贵,陈卿：微分几何,北京：高等教育出版社,2002,115-242
[7]S.S.Chem,W.H.Chen,K.S.Lam:Lectures on Differential geometry.2006,133-162
[8]Gursky,M.J., Locally conformally flat 4 and 6 manifolds of positive scalar curvature and positive Euler characteristic.Indiana Univ.Math.J.,43,1994,747-774
[9]Patodi,V.K.Curvature and the fundamental solutions of the heat operator. J.Diff. Geom.,5,1971,233-249
[10]Rosenberg, S. The variation of the de Rhamzeta function Trans. Amer. Math. Soc.,299(2), 1987,535-557
[11]Patodi,V.K.Curvature and the fundamental solutions of the heat operator. J.Diff. Geom., 5,1971,233-249
[12]T.E.Cecil,P.T.Ryan Tight and Taut Immersions of Manifolds,Boston,Pitman,1985
[13]Sylvestre Gallot Dominique Hulin Jacques Lafontaine. Riemannian Geometry 3rd ed.2008, 129-151
[14]K.Ueno.Some new examples of eigenmaps from Sm into Sn,Proc.Japan Acad.Ser.A 6,1993,205-208
[15]Y.L.Ou. Generalized Hopf constructions and eigeenmaps between spheres,Contributions to Algebra and Geomrtry,40,1999,267-274
[16]G.Toth.Harmonic Maps and Minimal Immersions Through Representation Theory, Academic Press,Boston,1990
[17]Jurgen Jost.Riemannian Geometry and Geometric Analysis 4th ed,2008,105-169
[18]H.Gauchman,G.Toth.Constructions of harmonic polynomial maps between spheres, Geom.Dedicata 50 1994,57-79
[19]Eells J, and Lemarie L. A Harmonic maps and minimal immersions with symmetries [M].New Jersey:Ann.Math.Stud.No130,Princeton Univ. Press,1993
[20]Baurguignon J.P.:An Introduction to Geometric Variational Problems, in Lectures on Geometric Variational Problems, Nishikawa R.Schoen(Eds),Springer-Verlag,Tokyo,1996
[21]P.Yiu.Quadratic forms between spheres and the non-existence of sums of squares formulae,Math.Proc.Camb.Phil.Soc.100,1986,493-504
[22]Cao,L.F.,Li,H.Z.r-Minimal Submanifolds in space forms. Ann.Glob. Anal. Geom,32(4), 2007,311-341
[23]J.F.Adams. Vector fields on spheres, Ann of Math.75,1962,603-632
[24]J.Adem.On the Hurwitz problem over arbitrary fieldsI,Bol.Soc.Mat.Mexicana,1980
[25]J.W.S.Cassels.On the representation of rational functions as sums of squares, Acta Arith.4,1964,79-82
[26]J.Eells.L.LemaireSelected topics in harmonic maps, CBMS Regional Conf.Ser.in Math.50 Amer.Math.Soc.,Providence,RI,1983
[27]Reilly,R.C. Variational properties of functions of the mean curvatures for hypersurfaces in space forms.J.Diff.Geom.,8(3),1973,465-477
[28]Berger,M.Riemannian Geometry during the second half of the twentieth century, University Lecture Series,Vol.17,AMS,Providence,Rhode Island,2000
[29]Gursky, M.J.,ViaclovskyJ.A. A new variational characterization of three-dimensional space forms.Invent.Math.,145,2001,251-278
[30]J.Levine.Embedding and immersion of real projective spaces, Proc. Amer.Math.Soc.14, 1963,801-803
[31]D.B.Shapiro.Compositions of quadratic forms, de Gruyter Exp.Math.33,Walter de Gruyter &Co.Berlin,2000
[32]Muto.Y. On Einstein metrics.J.Diff.Geom.9,1974,521-530
[33]Besse,A.L. Einstein Manifolds,Springer-Verlag, Berlin,1987,92-212
[34]Toth G.On classification of orthogonal multiplications a la Do-Carmo-Wallach[J].Gemo. Dedicata.1987,251-254
[2]Hu,Z.J.,Li H.Z. A new variational characterization of n-dimensional space forms. Trans. Amer. Math.Soc.,356(8),2004,3005-3023
[3]Fa En WU.On the Variation of a metric and its Application. The Editorial Office of AMS and Springer-Verlag,2009
[4]老大中：变分法基础(第2版),北京：国防工业出版社,2006,45-96
[5]陈维桓,李兴校：黎曼儿何引论上册,北京：北京大学出版社2002,171-257
[6]彭家贵,陈卿：微分几何,北京：高等教育出版社,2002,115-242
[7]S.S.Chem,W.H.Chen,K.S.Lam:Lectures on Differential geometry.2006,133-162
[8]Gursky,M.J., Locally conformally flat 4 and 6 manifolds of positive scalar curvature and positive Euler characteristic.Indiana Univ.Math.J.,43,1994,747-774
[9]Patodi,V.K.Curvature and the fundamental solutions of the heat operator. J.Diff. Geom.,5,1971,233-249
[10]Rosenberg, S. The variation of the de Rhamzeta function Trans. Amer. Math. Soc.,299(2), 1987,535-557
[11]Patodi,V.K.Curvature and the fundamental solutions of the heat operator. J.Diff. Geom., 5,1971,233-249
[12]T.E.Cecil,P.T.Ryan Tight and Taut Immersions of Manifolds,Boston,Pitman,1985
[13]Sylvestre Gallot Dominique Hulin Jacques Lafontaine. Riemannian Geometry 3rd ed.2008, 129-151
[14]K.Ueno.Some new examples of eigenmaps from Sm into Sn,Proc.Japan Acad.Ser.A 6,1993,205-208
[15]Y.L.Ou. Generalized Hopf constructions and eigeenmaps between spheres,Contributions to Algebra and Geomrtry,40,1999,267-274
[16]G.Toth.Harmonic Maps and Minimal Immersions Through Representation Theory, Academic Press,Boston,1990
[17]Jurgen Jost.Riemannian Geometry and Geometric Analysis 4th ed,2008,105-169
[18]H.Gauchman,G.Toth.Constructions of harmonic polynomial maps between spheres, Geom.Dedicata 50 1994,57-79
[19]Eells J, and Lemarie L. A Harmonic maps and minimal immersions with symmetries [M].New Jersey:Ann.Math.Stud.No130,Princeton Univ. Press,1993
[20]Baurguignon J.P.:An Introduction to Geometric Variational Problems, in Lectures on Geometric Variational Problems, Nishikawa R.Schoen(Eds),Springer-Verlag,Tokyo,1996
[21]P.Yiu.Quadratic forms between spheres and the non-existence of sums of squares formulae,Math.Proc.Camb.Phil.Soc.100,1986,493-504
[22]Cao,L.F.,Li,H.Z.r-Minimal Submanifolds in space forms. Ann.Glob. Anal. Geom,32(4), 2007,311-341
[23]J.F.Adams. Vector fields on spheres, Ann of Math.75,1962,603-632
[24]J.Adem.On the Hurwitz problem over arbitrary fieldsI,Bol.Soc.Mat.Mexicana,1980
[25]J.W.S.Cassels.On the representation of rational functions as sums of squares, Acta Arith.4,1964,79-82
[26]J.Eells.L.LemaireSelected topics in harmonic maps, CBMS Regional Conf.Ser.in Math.50 Amer.Math.Soc.,Providence,RI,1983
[27]Reilly,R.C. Variational properties of functions of the mean curvatures for hypersurfaces in space forms.J.Diff.Geom.,8(3),1973,465-477
[28]Berger,M.Riemannian Geometry during the second half of the twentieth century, University Lecture Series,Vol.17,AMS,Providence,Rhode Island,2000
[29]Gursky, M.J.,ViaclovskyJ.A. A new variational characterization of three-dimensional space forms.Invent.Math.,145,2001,251-278
[30]J.Levine.Embedding and immersion of real projective spaces, Proc. Amer.Math.Soc.14, 1963,801-803
[31]D.B.Shapiro.Compositions of quadratic forms, de Gruyter Exp.Math.33,Walter de Gruyter &Co.Berlin,2000
[32]Muto.Y. On Einstein metrics.J.Diff.Geom.9,1974,521-530
[33]Besse,A.L. Einstein Manifolds,Springer-Verlag, Berlin,1987,92-212
[34]Toth G.On classification of orthogonal multiplications a la Do-Carmo-Wallach[J].Gemo. Dedicata.1987,251-254