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给定的n维黎曼流形(M,g)能否等距浸入或嵌入到N维欧氏空间中是微分几何的经典问题。本文对黎曼流形等距浸入和嵌入问题进行了系统梳理,首先介绍了该问题的研究背景、发展脉络、基本研究方法以及经典结果等,然后阐述了在不同高斯曲率情况下的研究进展,最后对未来研究方向进行前瞻性展望。
Abstract:Whether a given n-dimensional Riemannian manifold(M,g) can be isometrically immersed or embedded into an N-dimensional Euclidean space is a classic problem in differential geometry. In this article, we provide a systematic review of isometric immersion and embedding problems in Riemannian manifolds. Firstly, the research background, development trajectory, basic research methods, and classical results of this problem are introduced. Then, the research progress under different Gaussian curvatures is elaborated. Finally, a forward-looking perspective is made on future research directions.
[1] Gauss C F.Disquisitiones generales circa superficies curvas[J].Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores,1828,6:99-146.
[2] Codazzi D.Sulle coordinate curvilinee d’una superficie dello spazio[J].Annali di Matematica Pura ed Applicata,1868,2:101-119.
[3] Darboux J G.Lectures on the general theory of surfaces (Vols.1-4)[M].Mineola,NY:Dover Publications,1972.
[4] Chen P,Wang M,Yau S.Conserved quantities in general relativity:from the quasi-local level to spatial infinity[J].Communications in Mathematical Physics,2015,338:31-80.
[5] Chen D,Tang Z,Xu Z,et al.Gaussian fusion:Accurate 3D reconstruction via geometry-guided displacement interpolation[J].2021 IEEE/CVF International Conference on Computer Vision (ICCV),Montreal,QC,Canada,2021:5896-5905.
[6] Zoltán M,Balogh T,Dániel V.Isometries and isometric embeddings of Wasserstein spaces over the Heisenberg group[EB/OL].(2023-05-09)[2025-02-24].https://arxiv.org/abs/2303.15095.
[7] Chen G Q,Slemrod M,Wang D H.Fluids,geometry,and the onset of Navier-Stokes turbulence in three space dimensions[J].Physica D:Nonlinear Phenomena,2018,376-377:23-30.
[8] Vaziri A,Mahedevan L.Localized and extended deformations of elastic shells[J].Proceedings of the National Academy of Sciences,USA ,2008,105:7913-7918.
[9] Schl?fli L.Nota alla memoria del.Sig.Beltrami,Sugli spazii di curvatura constante[J].Annali di Matematica Pura ed Applicata,1871-1873,5:170-193.
[10] Hilbert D.Ueber fl?chen von constanter Gaussscher krümmung[J].Transactions of the American Mathematical Society,1901,2:87-99.
[11] Tompkins C.Isometric embedding of flat manifolds in Euclidean space[J].Duke Mathematical Journal,1939,5:58-61.
[12] Chern S S,Kuiper N H.Some theorems on the isometric embedding of compact Riemannian manifolds in Euclidean space[J].Annals of Mathematics,1952,56:422-430.
[13] Janet M.Sur la possibilité de plonger un espace Riemannian donné dans un espace Euclidien[J].Annales de la societe polonaise de mathematique,1926,5:38-43.
[14] Cartan E.Sur la possibilité de plonger un espace Riemannian donné dans un espace Euclidien[J].Annales de la societe polonaise de mathematique,1927,6:1-7.
[15] 王萼芳,石生明.高等代数(第五版)[M].北京:高等教育出版社,2019.
[16] 陈省身,陈维桓.微分几何讲义(第二版)[M].北京:北京大学出版社,2001.
[17] Nirenberg L.The Weyl and Minkowski problems in differential geometry in the large[J].Communications on Pure and Applied Mathematics,1953,6:337-394.
[18] Guan P,Li Y.The Weyl problem with nonnegative Gauss curvature[J].Journal of Differential Geometry,1994,39:331-342.
[19] Hilbert D,Cohn V S.Geometry and the Imagination[M].New York:Chelsea Publishing Company,1952.
[20] Carmo M P.Riemannian Geometry.Translated by Francis Flaherty[M].Boston,MA:Mathematics:Theory and Applications,1992.
[21] Burago Y D,Shefel S Z.The geometry of surfaces in Euclidean spaces,Geometry III:Theory of Surfaces,edited by Yu.D.Burago and V.A.Zalgaller[M].Berlin Heidelberg:Springer,1992,1-85.
[22] Eisenhart L P.Riemannian Geometry.Eighth Printing[M].Princeton,New Jersey:Princeton University Press,1997.
[23] Mardare S.The foundamental theorem of theorey for surfaces with little regularity[J].Journal of Elasticity,2003,73(1):251-290.
[24] Poznyak E G,Shikin E V.Small parameters in the theory of isometric imbeddings of two dimensional Riemannian manifolds in Euclidean spaces.In:Some Questions of Differential Geometry in the Large[J].American Mathematical Society Translations,Series 2,1996,176:151-192.
[25] Cao W,Han Q,Huang F,et al.The isometric immersion of surfaces with finite total curvature[EB/OL].(2023-08-05)[2025-02-20].https://arxiv.org/abs/2308.02832.
[26] Chen G Q,Clelland J,Slemrod M,et al.Isometric embedding via strongly symmetric positive systems[J].Asian Journal of Mathematics,2018,22:1-40.
[27] Han Q,Hong J X.Isometric embedding of Riemannian manifolds in Euclidean spaces[M].Providence,Rhode Island:American Mathematical Society,2006.
[28] Chen G Q,Slemrod M,Wang D H.Weak continuity of the Gauss-Codazzi-Ricci system for isometric embedding[J].Proceedings of the American Mathematical Society,2010,138:1843-1852.
[29] Chen G Q,Slemrod M,Wang D H.Isometric immersions and compensated compactness[J].Communications in Mathematical Physics,2010,294:411-437.
[30] Evans L C.Weak Convergence Methods for Nonlinear Partial Differential Equations[M].Providence,Rhode Island:American Mathematical Society,1990.
[31] Weyl H.Uber die Bestimmung einer geschlossen konvexen Flache durch ihr Linienelement[J].Vierteljahrsschrift der naturforschenden Gesellschaft,1916,61:40-72.
[32] Aleksandrov A D.Vnutrennyaya Geometriya Vypuklyh Poverhnostei (in Russian)[M].Moscow:Gos.Izd-vo Techniko-Teoreticheskoj Literatury,1948.
[33] Nash J.C1isometric imbeddings[J].Annals of Mathematics,1954,60:383-396.
[34] Nash J.The imbedding problem for Riemannian manifolds[J].Annals of Mathematics,1956,63:20-63.
[35] Kuiper N.On C1isometric embeddings I,II[J].Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen,1955,58:545-556,683-689.
[36] Pogorelov A V.Extrinsic geometry of convex surfaces[M].Providence,Rhode Island:American Mathematical Society,1973.
[37] Tenenblat K.On isometric immersions of Riemannian manifolds[J].Bulletin of the Brazilian Mathematical Society,1971,2:23-36.
[38] Wen G H,Jiang L,Wen J.Local relative transformation with application to isometric embedding[J].Pattern Recognition Letters,2009,30(3):203-211.
[39] Ghomi M,Greene R E.Relative isometric embeddings of riemannian manifolds[J].Transactions of the American Mathematical Society,2011,363(1):63-73.
[40] Saucan E.Isometric embeddings in imaging and vision:facts and fiction[J].Journal of Mathematical Imaging And Vision,2012,43(2):143-155.
[41] Acharya A,Chen G Q,Li S R,et al.Fluids,elasticity,geometry,and the existence of wrinkled solutions[J].Archive:Rational Mechanics and Analysis,2017,226:1009-1060.
[42] Lewy H.On the existence of a closed convex surface realizing a given riemannian metric[J].Proceedings of the National Academy of Sciences of the United States of America,1938,24(2):104-106.
[43] Alexandroff A D.Existence of a convex polyhedron and of a convex surface with a given metric[J].Mathematical Collection,1942,53(1):15-65.
[44] Pogorelov A V.Regularity of a convex surface with given Gaussian curvature[J].Mathematical Collection,1952,73(1):88-103.
[45] Hong J X,Zuily C.Isometric embedding of the 2-sphere with non negative curvature in R3[J].Mathematische Zeitschrift,1995,219(1):323-334.
[46] Cohn V.Zwei satze uber die starrheit der eiflachen[J].Mathematical Physics,1927,1927:12-134.
[47] Herglotz G.Uber die starrheit der eiflachen[J].Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg,1943,15(1):127-129.
[48] Blaschke W.Vorlesungen uber differential geometrie I[J].Monatshefte Fur Mathematik Und Physik,1945,23(1):A12-A13.
[49] Efimov N V.The impossibility in Euclidean 3-space of a complete regular surface with a negative upper bound of the Gaussian curvature[J].Doklady Akademii Nauk SSSR,1963,150:1206-1209.
[50] Wehrheim K.Uhlenbeck compactness[M].Zürich:EMS Series of Lectures in Mathematics,2003.
[51] Guan P,Lu S.Curvature estimates for immersed hypersurfaces in Riemannian manifolds[J].Inventiones Mathematicae,2017,208 :191-215.
[52] Li S R.The Weyl problem of isometric immersions revisited[J].Bulletin of the London Mathematical Society,2021,53:220-230.
[53] Isobe T,Xu T T.Solutions of spinorial Yamabe-type problems on Sm:Perturbations and Applications[J].Transactions of the American Mathematical Society,2023,376:6397-6446.
[54] Lin C S.The local isometric embedding in R3 of 2-dimensional Riemannian manifolds with Gaussian curvature changing sign cleanly[J].Communications on Pure and Applied Mathematics,1986,39:867 -887.
[55] Han Q.On isometric embedding of surfaces with Gauss curvature changing sign cleanly[J].Communications on Pure and Applied Mathematics,2005,58:285-295.
[56] Han Q.Local isometric embedding of surfaces with Gauss curvature changing sign stably across a curve[J].Calculus of Variations and Partic Differential Equations,2006,25(1):79-103.
[57] Cao W T.The semi-global isometric embedding of surfaces with curvature changing signs stably[J].Proceedings of the American Mathematical Society,2019,147:4343-4353.
[58] Hong J.Realization in R3 of complete Riemannian manifolds with negative curvature[J].Communications in Analysis Geometry,1993,1(4):487-514.
[59] Li S R.Some recent developments on isometric immersions via compensated compactness and gauge transforms[J].Communications in Mathematical Analysis and Applications,2024,3(4):532-557.
[60] Christoforou C.BV weak solutions to Gauss-Codazzi system for isometric immersions[J].Journal of Differential Equations,2012,252(3):2845-2863.
[61] Cao W T,Huang F M,Wang D H.Isometric immersions of surfaces with two classes of metrics and negative Gauss curvature[J].Archive for Rational Mechanics and Analysis,2015,218:1431-1457.
[62] Cao W T,Huang F M,Wang D H.Isometric immersion of surface with negative Gauss curvature and the Lax-Friedrichs scheme[J].SIAM Journal on Mathematical Analysis,2016,48:2227-2249.
[63] Li S R.On the existence of C1,1-isometric immersions of several classes of negatively curved surfaces into R3[J].Archive for Rational Mechanics and Analysis,2020,236:419-449.
[64] Mardare S.On Pfaff systems with Lpcoefficients and their applications in differential geometry[J].Journal de Mathematiques Pures et Appliquees,2005,84:1659-1692.
[65] Mardare S.On systems of first order linear partial differential equations with Lpcoefficients[J].Advances in Differential Equations,2007,12:301-360.
[66] Chen G Q,Chen G,Li S R.Global weak rigidity of the Gauss-Codazzi-Ricci equations and isometric immersions of Riemannian manifolds with lower regularity[J].Journal of Geometric Analysis,2018,28:1957-2007.
[67] Ciarlet P G,Larsonneur F.On the recovery of a surface with prescribed first and second fundamental forms[J].Journal de Mathématiques Pures et Appliquées,2002,81:167-185.
[68] Ciarlet P G,Mardare C.A surface in W2,pis a locally Lipschitz-continuous function of its fundamental forms in W1,pand Lp,p>2[J].Journal de Mathématiques Pures et Appliquées,2019,124:300-318.
[69] Szopos M.An existence and uniqueness result for isometric immersions with little regularity[J].Revue Roumaine de Mathématiques Pures et Appliquées,2008,53:555-565.
[70] Chen G Q,Chen G,Li S R.Weak continuity of the Cartan structural system and compensated compactness on semi-Riemannian manifolds with lower regularity[J].Archive for Rational Mechanics and Analysis,2021,241:579-641.
[71] Chen G Q,Li S R,Slemrod M.On asymptotic rigidity and continuity problems in nonlinear elasticity onmanifolds and hypersurfaces[J].Journal de Mathématiques Pures et Appliquées,2022,160:29-53.
[72] Li S,Su X.On the fundamental theorem of submanifold theory and isometric immersions with super critical low regularity[EB/OL].(2024-05-25)[2025-02-25].https://arxiv.org/abs/2405.16249.
[73] Morrey C B.The problem of Plateau on a Riemannian manifold[J].Annals of Mathematics,1948,49:807-851.
[74] Litzinger F.Optimal regularity for two-dimensional Pfaffian systems and the fundamental theorem of surface theory[J].Journalof Geometric Analysis,2021,31:2594-2610.
[75] Berger E,Bryant R,Griffiths P.Some isometric embedding and rigidity results for Riemannian manifolds[J].Proceedings of the National Academy of Sciences of the United States of America,1981,78:4657-4660.
[76] Berger E,Bryant R,Griffiths P.Characteristics and rigidity of isometric embeddings[J].Duke Mathematical Journal,1983,50:803-892.
[77] Bryant R L,Griffiths P A,Yang D.Characteristics and existence of isometric embeddings[J].Duke Mathematical Journal,1983,50:893-994.
[78] Cao W T,Székelyhidi L.C1,αisometric extensions[J].Communications in Partial Differential Equations,2019,44(7):613-636.
[79] Cao W T,Székelyhidi L.Global Nash-Kuiper theorem for compact manifolds[J].Journal of Differential Geometry,2022,122 (1) :35 - 68.
[80] Cao W T,Inauen D.Rigidity and flexibility of isometric extensions[J].Commentarii Mathematici Helvetici,2024,99(1):39-80.
[81] Cao W T,Székelyhidi L.On the isometric version of Whitney′s strong embedding theorem[J].Advances in Mathematics,2025,460:110040.
[82] Gromov M.Partial Differential Relations[M].Berlin,Heidelberg:Springer,1986.
[83] Nakamura G,Maeda Y.Local isometric embedding problem of Riemannian 3-manifold into R6[J].Proceedings of the Japan Academy.Series A,Mathematical Sciences,1986,62 (7):257-259.
[84] Nakamura G,Maeda Y.Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces[J].Transactions of the American Mathematical Society,1989,313 (1):1-51.
[85] Poole T E.The local isometric embedding problem for 3-dimensional Riemannian manifolds with cleanly vanishing curvature[J].Communications in Partial Differential Equations,2010,35:1802-1826.
[86] Wang X,Zhu K.Isometric embeddings via heat kernel[J].Journal of Differential Geometry,2015,99(3):497-538.
[87] Li S R,Slemrod M.From the Nash - Kuiper theorem of isometric embeddings to the Euler equations for steady fluid motions:analogues,examples,and extensions[J].Journal of Mathematical Physics,2023,64(1):011511.
[88] Lewicka M,Pakzad M.Convex integration for the Monge-Ampère equation in two dimensions[J].Analysis and Partial Differential Equation,2017,10(3):695-727.
[89] Cao W T,Wang Z H.III-posedness of the Dirichlet problem for 2D Lagrangian mean curvature equation[EB/OL].(2024-09-07)[2025-02-25].https://arxiv.org/abs/2409.04816.
基本信息:
DOI:
中图分类号:O186.12
引用信息:
[1]房茜茜,于慧敏.等距浸入或嵌入问题的相关研究[J].山东师范大学学报(自然科学版),2025,40(03):232-246.
基金信息:
国家自然科学基金资助项目(12271310)