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Riemannian Submanifolds: a Survey

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RIEMANNIAN SUBMANIFOLDS: A SURVEY BANG-YEN CHEN Contents Chapter 1. Introduction .............................. ...................6 Chapter 2. Nash’s embedding theorem and some related results .........9 2.1. Cartan-Janet’s theorem .......................... ...............10 2.2. Nash’s embedding theorem ......................... .............11 2.3. Isometric immersions with the smallest possible codimension . 8 2.4. Isometric immersions with prescribed Gaussian or Gauss-Kronecker curvature .......................................... ..................12 2.5. Isometric immersions with prescribed mean curvature. ...........13 Chapter 3. Fundamental theorems, basic notions and results ...........14 3.1. Fundamental equations ........................... ..............14 3.2. Fundamental theorems ............................ ..............15 3.3. Basic notions ................................... ................16 3.4. A general inequality ............................. ...............17 3.5. Product immersions .............................. .............. 19 3.6. A relationship between k-Ricci tensor and shape operator . 20 3.7. Completeness of curvature surfaces . ..............22 Chapter 4. Rigidity and reduction theorems . ..............24 4.1. Rigidity ....................................... .................24 4.2. A reduction theorem .............................. ..............25 Chapter 5. Minimal submanifolds ....................... ...............26 arXiv:1307.1875v1 [math.DG] 7 Jul 2013 5.1. First and second variational formulas . .............27 5.2. Jacobi operator, index, nullity and Killing nullity . .............28 5.3. Minimal submanifolds in Euclidean space . ...........28 5.4. Minimal submanifolds in sphere .................... .............41 5.5. Minimal submanifolds in hyperbolic space . ............49 5.6. Gauss map of minimal surfaces ...................... ...........50 Riemannian Submanifolds, Handbook of Differential Geometry, vol. I (2000), 187–418. 1 2 B.-Y. CHEN 5.7. Complete minimal submanifolds in Euclidean space with finite total curvature.......................................... ......... 54 5.8. Complete minimal surfaces in E3 lying between two parallel planes............................................. ..........67 5.9. The geometry of Gauss image ........................ ........... 67 5.10. Stability and index of minimal submanifolds . ............68 Chapter 6. Submanifolds of finite type ................... ..............77 6.1. Spectral resolution ............................. .................77 6.2. Order and type of immersions ....................... ............78 6.3. Equivariant submanifolds as minimal submanifolds in their adjoint hyperquadrics...................................... .........80 6.4. Submanifolds of finite type ........................ ..............80 Chapter 7. Isometric immersions between real space forms . ...........89 Chapter 8. Parallel submanifolds ...................... .................92 8.1. Parallel submanifolds in Euclidean space . .............92 8.2. Parallel submanifolds in spheres .................. ...............94 8.3. Parallel submanifolds in hyperbolic spaces . .............94 8.4. Parallel submanifolds in complex projective and complex hyperbolic spaces ............................................. .........94 8.5. Parallel submanifolds in quaternionic projective spaces ..........95 8.6. Parallel submanifolds in the Cayley plane . ............95 Chapter 9. Standard immersions and submanifolds with simple geodesics 96 9.1. Standard immersions ............................. ..............96 9.2. Submanifolds with planar geodesics . .............96 9.3. Submanifolds with pointwise planar normal sections . ...........97 9.4. Submanifolds with geodesic normal sections and helical immersions......................................... .........98 9.5. Submanifolds whose geodesics are generic W -curves .............99 9.6. Symmetric spaces in Euclidean space with simple geodesics . 101 Chapter 10. Hypersurfaces of real space forms ............ ............103 10.1. Einstein hypersurfaces ......................... ...............103 10.2. Homogeneous hypersurfaces ...................... ............103 10.3. Isoparametric hypersurfaces .................... ..............104 10.4. Dupin hypersurfaces ............................ ..............109 10.5. Hypersurfaces with constant mean curvature . ..........117 10.6. Hypersurfaces with constant higher order mean curvature .....123 10.7. Harmonic spaces and Lichnerowicz conjecture . ..........124 Chapter 11. Totally geodesic submanifolds . ..............127 11.1. Cartan’s theorem ............................... ..............127 RIEMANNIANSUBMANIFOLDS 3 11.2. Totally geodesic submanifolds of symmetric spaces . ..........127 11.3. Stability of totally geodesic submanifolds . .............133 11.4. Helgason’s spheres ............................. ...............136 11.5. Frankel’s theorem.............................. ..............1387 Chapter 12. Totally umbilical submanifolds . ..............139 12.1. Totally umbilical submanifolds of real space forms . ..........139 12.2. Totally umbilical submanifolds of complex space forms ........140 12.3. Totally umbilical submanifolds of quaternionic space forms ....140 12.4. Totally umbilical submanifolds of the Cayley plane . ..........140 12.5. Totally umbilical submanifolds in complex quadric . ..........141 12.6. Totally umbilical submanifolds of locally symmetric spaces . 141 12.7. Extrinsic spheres in locally symmetric spaces . ...........142 12.8. Totally umbilical hypersurfaces . ..............143 Chapter 13. Conformally flat submanifolds ............... .............145 13.1. Conformally flat hypersurfaces ................... .............145 13.2. Conformally flat submanifolds .................... ............147 Chapter 14. Submanifolds with parallel mean curvature vector ........151 14.1. Gauss map and mean curvature vector . ........ 151 14.2. Riemann sphere with parallel mean curvature vector . ........151 14.3. Surfaces with parallel mean curvature vector . ...........152 14.4. Surfaces with parallel normalized mean curvature vector . 153 14.5. Submanifolds satisfying additional conditions . .............153 14.6. Homogeneous submanifolds with parallel mean curvature vector 154 Chapter 15. K¨ahler submanifolds of K¨ahler manifolds . .............156 15.1. Basic properties of K¨ahler submanifolds . ............156 15.2. Complex space forms and Chern classes .............. .........157 15.3. K¨ahler immersions of complex space forms in complex space forms .............................................. ....... 158 15.4. Einstein-K¨ahler submanifolds and K¨ahler submanifolds satisfying Ric(X,Y )= Ric(X,Y ) ....................................159 15.5. Ogiue’s conjectures and curvature pinching . ...........160 15.6. Segre embeddinge ................................ .............162 15.7. Parallel K¨ahler submanifolds . ..............162 15.8. Symmetric and homogeneous K¨ahler submanifolds . ........163 15.9. Relative nullity of K¨ahler submanifolds and reduction theorems 164 Chapter 16. Totally real and Lagrangian submanifolds of K¨ahler manifolds .......................................... ....... 165 16.1. Basic properties of Lagrangian submanifolds . ...........165 16.2. A vanishing theorem and its applications . ..........167 4 B.-Y. CHEN 16.3. The Hopf lift of Lagrangian submanifolds of nonflat complex space forms ......................................... 168 16.4. Totally real minimal submanifolds of complex space forms ....170 16.5. Lagrangian real space form in complex space form . .......171 16.6. Inequalities for Lagrangian submanifolds . .............172 16.7. Riemannian and topological obstructions to Lagrangian immersions ......................................... .......173 16.8. An inequality between scalar curvature and mean curvature . .174 16.9. Characterizations of parallel Lagrangian submanifolds ........177 16.10. Lagrangian H-umbilical submanifolds and Lagrangian catenoid 137 16.11. Stability of Lagrangian submanifolds . ..........178 16.12. Lagrangian immersions and Maslov class . .......179 Chapter 17. CR-submanifolds of K¨ahler manifolds .................... 181 17.1. Basic properties of CR-submanifolds of K¨ahler manifolds . 181 17.2. Totally umbilical CR-submanifolds ...........................182 17.3. Inequalities for CR-submanifolds .............................183 17.4. CR-products .......................................... .......184 17.5. Cyclic parallel CR-submanifolds ..............................185 17.6. Homogeneous and mixed foliate CR-submanifolds .............186 17.7. Nullity of CR-submanifolds ...................................186 Chapter 18. Slant submanifolds of K¨ahler manifolds . .............188 18.1. Basic properties of slant submanifolds . ............188 18.2. Equivariant slant immersions . .............189 18.3. Slant surfaces in complex space forms . ...........190 18.4. Slant surfaces and almost complex structures . ..........192 18.5. Slant spheres in complex projective spaces . ...........193 Chapter 19. Submanifolds of the nearly K¨ahler 6-sphere . ...........196 19.1. Almost complex curves ........................... ............196 19.2. Minimal surfaces of constant curvature in the nearly K¨ahler 6-sphere ........................................... ........199 19.3. Hopf hypersurfaces and almost complex curves . .........200 19.4. Lagrangian submanifolds in nearly K¨ahler 6-sphere . ...........200
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