Differential geometry : proceedings of the VIII International Colloquium, Santiago de Compostela, Spain, 7-11 July 2008

This volume contains research and expository papers on recent advances in foliations and Riemannian geometry. Some of the topics covered in this volume include: topology, geometry, dynamics and analysis of foliations, curvature, submanifold theory, Lie groups and harmonic maps. Among the contributions, readers may find an extensive survey on characteristic classes of Riemannian foliations offering also new results, an article showing the uniform simplicity of certain diffeomorphism groups, an exposition of convergences of contact structures to foliations from the point of view of Thurston's and Thurston-Bennequin's inequalities, a discussion about Fatou-Julia Decompositions for foliations and a description of singular Riemannian foliations on spaces without conjugate points. Papers on submanifold theory focus on the existence of graphs with prescribed mean curvature and mean curvature flow for spacelike graphs, isometric and conformal deformations and detailed surveys on totally geodesic submanifolds in symmetric spaces, cohomogeneity one actions on hyperbolic spaces and rigidity of geodesic spheres in space forms. Geometric realizability of curvature tensors and curvature operators are also treated in this volume with special attention to the affine and the pseudo-Riemannian settings. Also, some contributions on biharmonic maps and submanifolds enrich the scope of this volume in providing an overview of different topics of current interest in differential geometry. CONTENTS......Page 10 Preface......Page 6 Organizing Committees......Page 8 A brief portrait of the life and work of Professor Enrique Vidal Abascal L. A. Cordero......Page 14 Part A Foliation theory......Page 22 1. Introduction......Page 24 2. Classifying spaces......Page 26 3. Primary classes......Page 29 4. Secondary classes......Page 34 5. Variation of secondary classes......Page 40 6. Molino Structure Theory......Page 43 7. Some open problems......Page 45 References......Page 46 1. Introduction......Page 49 2. The construction......Page 51 3. Questions......Page 54 References......Page 55 1. Introduction......Page 56 2. Uniform perfectness of diffeomorphism groups......Page 59 3. Uniform simplicity of the diffeomorphism groups......Page 64 References......Page 68 1. Thurston's Inequality......Page 69 2. Thurston's Absolute Inequality for Spinnable Foliations......Page 72 3. Dehn Filling......Page 73 4. Bennequin's Isotopy Lemma......Page 74 References......Page 76 1. Introduction......Page 78 2. Definition of the Fatou and Julia sets......Page 79 3. Some properties of Julia sets......Page 83 4. Examples......Page 84 References......Page 86 1. Introduction......Page 88 2. Preliminaries......Page 90 3. Geometric arguments......Page 92 4. Topological arguments......Page 93 References......Page 94 0. Introduction......Page 96 1. Main Results......Page 98 2.1. Algebraic preliminaries......Page 100 2.3. Variational formulae......Page 102 References......Page 106 1. Introduction......Page 107 2. Canonical differential operators defined on a Riemannian foliation......Page 108 3. Adiabatic limits and Riemannian foliations......Page 110 4. A transversal Weitzenböck formula......Page 112 References......Page 114 1. The duality......Page 115 References......Page 116 Open problems on foliations......Page 117 Part B Riemannian geometry......Page 122 1. Introduction......Page 124 2. Killing graphs......Page 125 3. Riemannian submersions......Page 126 4. Conformal Killing graphs......Page 128 References......Page 132 1. Introduction......Page 133 2.1. Local theory for the isometric case......Page 134 2.3. Digression: Extending intrinsic isometries......Page 135 3. Higher codimensions: Rigidity results......Page 136 3.1. The s-nullity and the conformal s-nullity......Page 137 3.2. A main tool: flat bilinear forms......Page 138 3.3. Conformal geometry in the light cone......Page 139 4. The general deformation problem......Page 140 5. Genuine deformations......Page 141 5.2. Genuine conformal deformations of submanifolds......Page 143 5.3. Constructing conformal pairs from isometric ones......Page 145 References......Page 147 1. Totally geodesic submanifolds......Page 149 2. Maximal totally geodesic submanifolds in the Riemannian symmetric spaces of rank 2......Page 151 2.1. G +2 (Rn+2)......Page 152 2.2. G2 (Cn+2)......Page 154 2.3. G (Hn+2)......Page 155 2.7. E6=(U(1) Spin(10))......Page 156 2.12. G2......Page 157 References......Page 158 The orbits of cohomogeneity one actions on complex hyperbolic spaces J. C. Díaz-Ramos......Page 159 1. The geometry of the orbits......Page 160 2. Hypersurfaces with constant principal curvatures......Page 167 References......Page 168 1. Introduction......Page 169 2. Preliminaries......Page 170 3. Proof of Theorem 1.1......Page 172 4. Rigidity results in the Euclidean space......Page 175 References......Page 176 1. Introduction......Page 177 2. Bernstein-Calabi and Heinz-Chern type results......Page 179 3. The mean curvature flow......Page 181 4. Homotopy to a constant map......Page 185 Acknowledgements......Page 186 References......Page 187 1. Introduction......Page 188 1.2. Osserman geometry......Page 189 1.3. Affine geometry......Page 190 1.4. Torsion free connections and Riemannian geometry......Page 191 2. The proof of Theorem 1.3......Page 192 References......Page 197 1. Introduction......Page 198 2. Preliminaries......Page 199 3. Conformally Osserman multiply warped products......Page 200 4. Locally conformally at multiply warped products......Page 201 6. Multiply warped products of constant curvature......Page 204 References......Page 207 1. Riemannian reductive homogeneous spaces......Page 208 2.1. -symmetric spaces......Page 209 2.3. Riemannian and Indefinite Riemannian -symmetric spaces......Page 210 2.4. Irreducible Riemannian -symmetric spaces......Page 211 3. Classification of compact simple Z2 symmetric spaces......Page 213 4.1. Z2 -symmetric metrics on flag manifolds......Page 214 4.2. The Z2 -Riemannian symmetric space SO(2m)=Sp(m)......Page 216 References......Page 219 1. Introduction and preliminaries......Page 220 2. Preliminaries about H-type groups......Page 222 3.1. Constant osculating rank of the Jacobi operator along a special family of geodesics......Page 223 3.2. Resolution of the Jacobi equation......Page 225 3.3. Relation between both methods......Page 227 References......Page 229 1. Introduction......Page 230 2. Homogeneous geodesics in pseudo-Riemannian manifolds......Page 231 3. Homogeneous geodesics in affine manifolds......Page 232 5. G.o. manifolds of type A......Page 233 6. G.o. manifolds of type B......Page 236 7. General connection of type B......Page 238 References......Page 239 Conjugate connections and differential equations on infinite dimensional manifolds M. Aghasi, C. T. J. Dodson, G. N. Galanis and A. Suri......Page 240 1. Introduction......Page 241 2. Preliminaries......Page 242 3. Classification for vector bundle structures of T2M......Page 243 4. Connections and ordinary differential equations......Page 245 5. The Earle and Eells foliation theorem in Fréchet spaces......Page 248 References......Page 249 1. Introduction......Page 250 3. Totally biharmonic hypersurfaces......Page 251 4. Totally biharmonic surfaces of space forms......Page 254 5. Biharmonic curves in H3......Page 256 References......Page 258 1. Introduction......Page 260 2.1. Biharmonic maps......Page 261 2.2. The tangent bundle and the unit sphere bundle......Page 262 3. Homogeneous structures......Page 264 References......Page 268 1. Introduction......Page 270 2. Bihamonic maps and warped product manifolds......Page 271 3. Biharmonic submanifolds in space forms......Page 273 4. On the biharmonicity of the Gauss map......Page 276 References......Page 278 1. Introduction......Page 279 2. Preliminaries on contact pairs......Page 280 3.1. Almost contact structures......Page 281 3.2. Contact pair structures......Page 282 4. Compatible and associated metrics......Page 283 4.1. Orthogonal foliations......Page 287 References......Page 288 1. Introduction......Page 289 2. Paraquaternionic structures on manifolds......Page 290 3. Manifolds endowed with mixed 3-structures......Page 292 4. Normal semi-invariant submanifolds and mixed 3-structures......Page 295 Acknowledgements......Page 297 References......Page 298 2. An Extension of Myers' Theorem......Page 299 References......Page 303 1. Different types of Gray curvature conditions......Page 304 2. Gray curvature conditions for the Tanaka-Webster connection......Page 305 References......Page 307 1. Introduction......Page 309 2. Natural operators......Page 310 References......Page 313 1. Introduction......Page 314 2. The main result......Page 315 3. Proof of the main result......Page 317 References......Page 318 1. Introduction......Page 319 2. Geodesics on surfaces of revolution......Page 320 3. The Clairaut's relation......Page 322 References......Page 323 1. Introduction......Page 324 2. The quasi-constant holomorphic sectional curvatures of the cotangent bundles with general natural lifted metrics......Page 326 References......Page 328 1. Preliminaries......Page 329 2. Weingarten translation surfaces......Page 330 References......Page 333 1. Introduction and motivation......Page 334 2. The bundle of r-frames......Page 335 3. First order G-structures......Page 336 4. Second order G-structures......Page 337 References......Page 338 List of Participants......Page 340 A brief portrait of the life and work of Professor Enrique Vidal Abascal / L.A. Cordero -- pt. A. Foliation theory. Characteristic classes for Riemannian foliations / S. Hurder. Non unique-ergodicity of harmonic measures: Smoothing Samuel Petite's examples / B, Deroin. On the uniform simplicity of diffeomorphism groups / T. Tsuboi. On Bennequin's isotopy lemma and Thurston's inequality / Y. Mitsumatsu. On the Julia sets of complex codimension-one transversally holomorphic foliations / T. Asuke. Singular Riemannian foliations on spaces without conjugate points / A. Lytchak. Variational formulae for the total mean curvatures of a codimension-one distribution / V. Rovenski and P. Walczak. On a Weitzenböck-like formula for Riemannian foliations / V. Slesar. Duality and minimality for Riemannian foliations on open manifolds / X.M. Masa. Open problems on foliations -- pt. B. Riemannian geometry. Graphs with prescribed mean curvature / M. Dajczer. Genuine isometric and conformal deformations of submanifolds / R. Tojeiro. Totally geodesic submanifolds in Riemannian symmetric spaces / S. Klein. The orbits of cohomogeneity one actions on complex hyperbolic spaces / J.C. Díaz-Ramos. Rigidity results for geodesic spheres in space forms / J. Roth. Mean curvature flow and Bernstein-Calabi results for spacelike graphs / G. Li and I.M.C. Salavessa. Riemannian geometric realizations for Ricci tensors of generalized algebraic curvature operators / P. Gilkey, S. Nikc̮ević and D. Westerman. Conformally Osserman multiply warped product structures in the Riemannian setting / M. Brozos-Vázquez, M.E. Vázquez-Abal and R. Vázquez-Lorenzo. Riemannian [symbol]-symmetric spaces / M. Goze and E. Remm. Methods for solving the Jacobi equation. Constant osculating rank vs. constant Jacobi osculating rank / T. Arias-Marco. On the reparametrization of affine homogeneous geodesics / Z. Dus̮ek. Conjugate connections and differential equations on infinite dimensional manifolds / M. Aghasi [und weitere]. Totally biharmonic submanifolds / D. Impera and S. Montaldo. The biharmonicity of unit vector fields on the Poincaré half-space H[symbol] / M.K. Markellos. Perspectives on biharmonic maps and submanifolds / A. Balmus. Contact pair structures and associated metrics / G. Bande and A. Hadjar. Paraquaternionic manifolds and mixed 3-structures / S. Ianus and G.E. Vi̮lcu. On topological obstruction of compact positively Ricci curved manifolds / W.-H. Chen. Gray curvature conditions and the Tanaka-Webster connection / R. Mocanu. Riemannian structures on higher order frame bundles from classical linear connections / J. Kurek and W.M. Mikulski. Distributions on the cotangent bundle from torsion-free connections / J. Kurek and W.M. Mikulski. On the geodesics of the rotational surfaces in the Bianchi-Cartan-Vranceanu spaces / P. Piu and M.M. Profir. Cotangent bundles with general natural Kähler structures of quasi-constant holomorphic sectional curvatures / S.L. Druta̮. Polynomial translation Weingarten surfaces in 3-dimensional Euclidean space / M.I. Munteanu and A.I. Nistor. G-structures defined on pseudo-Riemannian manifolds / I. Sánchez-Rodríguez -- List of participants "This volume contains research and expository papers on recent advances in foliations and Riemannian geometry. Some of the topics covered in this volume include: topology, geometry, dynamics and analysis of foliations, curvature, submanifold theory, Lie groups and harmonic maps." "Among the contributions, readers may find an extensive survey on characteristic classes of Riemannian foliations offering also new results, an article showing the uniform simplicity of certain diffeomorphism groups, an exposition of convergences of contact structures to foliations from the point of view of Thurston's and Thurston-Bennequin's inequalities, a discussion about Fatou-Julia decompositions for foliations and a description of singular Riemannian foliations on spaces without conjugate points." "Papers on submanifold theory focus on the existence of graphs with prescribed mean curvature and mean curvature flow for spacelike graphs, isometric and conformal deformations and detailed surveys on totally geodesic submanifolds in symmetric spaces, cohomogeneity one actions on hyperbolic spaces and rigidity of geodesic spheres in space forms. Geometric realizability of curvature tensors and curvature operators are also treated in this volume with special attention to the affine and the pseudo-Riemannian settings. Also, some contributions on biharmonic maps and submanifolds enrich the scope of this volume in providing an overview of different topics of current interest in differential geometry."--Jacket