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Th`Ese De Doctorat Pour Obtenir Le Grade De Docteur De L'université De Th`esede doctorat Pour obtenir le grade de Docteur de l'Universit´ede VALENCIENNES ET DU HAINAUT-CAMBRESIS´ MATHEMATIQUES´ PURES Pr´esent´eeet soutenue par Marilena, Moruz Le 03/04/2017, `aValenciennes Ecole doctorale : Sciences Pour l'Ing´enieur(SPI) Equipe de recherche, Laboratoire : Laboratoire de Math´ematiques et ses Applications de Valenciennes (LAMAV) Etude des sous-vari´et´esdans les vari´et´esk¨ahl´eriennes, presque k¨ahl´erienneset les vari´et´esproduit JURY Pr´esidentdu jury Opozda, Barbara. Professeur. Jagiellonian University, Krakow, Poland. Rapporteurs Leschke, Katrin. Professeur. University of Leicester, Leicester, UK. Voisin, Claire. Professeur. Coll`egede France, Paris. Examinateurs Birembaux, Olivier. Professeur. Universit´ede Valenciennes. El Kacimi, Aziz. Professeur. Universit´ede Valenciennes. Munteanu, Marian Ioan. Professeur. UAIC University of Ia¸si,Roumanie. Van der Veken, Joeri. Professor. KU Leuven, Belgium. Directeur de th`ese Vrancken, Luc. Professeur. Universit´ede Valenciennes. 2 Contents R´esum´e 3 Summary 12 1 Preliminaries 21 1.1 Affine, Riemannian and semi-Riemannian manifolds . 21 1.1.1 Affine manifolds . 22 1.1.2 Riemannian manifolds . 22 1.1.3 Semi-Riemannian manifolds . 24 1.2 Submanifolds . 26 1.2.1 On affine hypersurfaces . 28 3 1.2.2 On surfaces in E and Lorentzian space . 31 3 3 2 On the Nearly K¨ahler S × S 34 3 3 2.1 Properties of the nearly K¨ahler S × S as a Riemannian submersion . 39 3 3 3 2.1.1 The structure on S × S × S ....................... 39 3 3 2.1.2 The nearly K¨ahler S × S as a Riemannian submersion . 40 2.1.3 Properties of the application F1, F2 ................... 43 2.1.4 The role of the almost product structure P ............... 46 2.2 Lagrangian submanifolds with constant angle functions in the nearly K¨ahler 3 3 S × S ....................................... 48 3 3 2.3 Lagrangian submanifolds of the nearly K¨ahler S × S from minimal surfaces 3 in S ......................................... 59 3 2.3.1 Elementary properties of orientable minimal surfaces in S . ....................................... 59 2.3.2 From the Lagrangian immersion to the minimal surface . 60 2.3.3 The reverse construction . 65 2.4 Conclusion . 81 3 A classification of isotropic affine hyperspheres 83 3.1 Possible dimensions and choice of frame . 86 3.2 Isotropic affine hyperspheres . 90 3.3 Affine hyperspheres of dimension 5 . 93 3.3.1 The form of L, dim U =1......................... 93 3.3.2 A canonical example . 94 3.4 Affine hyperspheres of dimension 8 . 96 3.4.1 The form of L, dim U =2......................... 96 1 3.4.2 Two canonical examples . 97 3.5 Affine hyperspheres of dimension 14 . 100 3.5.1 The form of L, dim U =4......................... 100 3.5.2 Two canonical examples . 102 3.6 Affine hyperspheres of dimension 26 . 106 3.6.1 The form of L, dim U =8......................... 106 3.6.2 Two canonical examples . 112 4 Translation and homothetical surfaces in Euclidean space with constant curvature 114 4.1 Proof of Theorem 49 . 118 4.1.1 Case K =0................................. 119 4.1.2 Case K 6=0................................. 119 4.2 Proof of Theorem 50 . 122 4.3 Proof of Theorem 51 . 123 4.3.1 Case K =0................................. 124 4.3.2 Case K 6=0................................. 124 4.4 The Lorentzian case . 126 R´esum´e Cette th`eseest constitu´eede quatre chapitres. Le premier contient les notions de base qui permettent d'aborder les divers th`emesqui y sont ´etudi´es. Le second est consacr´e`al'´etude 3 3 des sous-vari´et´eslagrangiennes d'une vari´et´epresque k¨ahl´erienne S × S . Une vari´et´epresque k¨ahl´erienneest une vari´et´epresque hermitienne munie d'une structure presque complexe J pour laquelle le tenseur (r~ J) est anti-sym´etrique,o`u r~ est la connexion 3 3 de Levi-Civita. Je m'int´eresse`ades sous-vari´et´eslagrangiennes de S × S non totalement 3 geod´esiquesdont la projection sur le premier facteur S n'est nulle part de rang maximal. J'exprime cette propri´et´e`al'aide des fonctions d'angle et j'´etudieplus particuli`erement la 3 relation entre ces sous-vari´et´eset les surfaces minimales dans S . Dans ce chapitre, je d´emontre 3 3 que les sous-vari´et´eslagrangiennes de S × S dont les fonctions d'angle sont constantes sont totalement g´eod´esiquesou ont leur courbure sectionnelle constante. Puis je donne une classification compl`etede ces sous-vari´et´eslagrangiennes. Les r´esultatspr´esent´esici ont ´et´e obtenus en collaboration avec Burcu Bekta¸s,Joeri Van der Veken et Luc Vrancken (voir [3] et [4]). Dans le troisi`emechapitre, je m'int´eresse`aun probl`emede g´eom´etrie diff´erentielle affine et donne une classification des hypersph`eresaffines qui sont isotropiques. Ce r´esultat a ´et´e obtenu en collaboration avec Luc Vrancken (voir [41]). Et enfin dans le dernier, je pr´esente quelques r´esultatssur les surfaces de translation et les surfaces homoth´etiques.Ce travail a ´et´er´ealis´eavec Rafael L´opez ([35]). Courte pr´esentation des r´esultatsobtenus 3 3 x L'´etudedes sous-vari´et´eslagrangiennes de la vari´et´epresque k¨ahl´erienne S ×S L'´etudedes vari´et´espresque k¨ahl´eriennesd´ebutedans les ann´ees1970 avec A. Gray. Des th´eor`emesde structure, traitant le cas tr`essp´ecialde la dimension 6, ont ´et´eobtenus par P.-A. Nagy dans les ann´ees2000. Plus r´ecemment, il a ´et´ed´emontr´epar J.-B. Butruille que 6 les seuls espaces presque k¨ahl´erienshomog`enesde dimension 6 sont la sph`ere S (avec la 7 3 3 structure presque complexe introduite par le produit vectoriel sur R ), l'espace S × S (mais pas ´equip´ede la m´etriquecanonique et muni d'une structure presque complexe introduite en 3 utilisant les quaternions), l'espace projectif CP (avec une m´etriqueet une structure presque complexe non canoniques) et la vari´et´edes drapeaux SU(3)=U(1) × U(1). Tous ces espaces sont compacts et 3-sym´etriques. En 2014 V. Cort´esand J. J. V´asquezont d´ecouvert les premi´eresstructures non homog`enes(mais locallement homog`enes)presque K¨ahlerdans [14]. 3 Les premi`eresstructures non homog`enescompl`etesdes espaces presque k¨ahl´eriensont ´et´e d´ecouvertes seulement en 2015 par L. Foscolo et M. Haskins (voir [23]). Il existe deux types de sous-vari´et´esdes espaces presque k¨ahl´eriens(ou plus g´en´eralement, presque hermitiens), `asavoir les sous-vari´et´espresque complexes et les sous-vari´et´estotalement r´eelles.Les sous- vari´et´espresque complexes sont des sous-vari´et´esdont les espaces tangents sont invariants par l'op´erateur J. Pour les sous-vari´et´estotalement r´eelles,les vecteurs tangents sont envoy´essur les vecteurs normaux par la structure presque complexe J. Dans ce dernier cas, si en plus la dimension de la sous-vari´et´eest la moiti´ede la dimension de l'espace ambiant, alors la sous- vari´et´eest appel´eelagrangienne. On notera que les sous-vari´et´eslagrangiennes des vari´et´es presque k¨ahl´eriennessont particuli`erement int´eressantes car elles sont toujours minimales et 6 orientables (voir [21] pour S et [27] ou [52] pour le cas g´en´eral). Les sous-vari´et´es lagrangiennes ont ´et´e´etudi´ees par de nombreux auteurs dans le pass´e. Par contre pour les autres espaces, jusqu’`apr´esent, il existe tr`espeu de r´esultats. Dans le 3 3 cas de S × S les premiers exemples ont ´et´eobtenus respectivement par Sch¨aferet Smoczyk (2010) et par Moroianu et Semmelmann (2014). Un exemple de tore plat et la classification de tous les exemples totalement g´eod´esiquesou avec courbure sectionelle constante ont ´et´e obtenus en 2014 par Dioos, Vrancken et Wang. Une phase fondamentale dans cette ´etudeest l'utilisation d'une structure presque produit 3 3 3 3 P sur S × S , qui est li´ee,mais est diff´erente de la structure produit canonique de S × S . La d´ecomposition de P en une partie tangentielle et une partie normale le long d'une sous- vari´et´elagrangienne permet alors d'introduire trois directions principales, E1, E2, E3, avec des fonctions angulaires correspondantes θ1, θ2, θ3. Les r´esultatsobtenus dans ce domaine sont pr´esent´esdans la suite. Theorem 1. Soit 3 3 f : M ! S × S x 7! f(x) = (p(x); q(x)) 3 une immersion Lagrangienne telle que la premi`ere projection p : M ! S est nulle part 3 une immersion (cela veut dire que p(M) est une surface dans R ). Alors une des fonctions π angulaires est constante et ´egale`a 3 . La r´eciproque est vraie aussi. Theorem 2. Soit 3 3 f : M ! S × S x 7! f(x) = (p(x); q(x)); 3 une immersion lagrangienne telle que la premi`ere projection p : M ! S est nulle part une 3 immersion. Alors p(M) est une surface minimale dans S . En distinguant plusieurs cas, nous avons aussi r´eussi`amontrer la r´eciproque, c’est-`a-dire 3 comment construire, `apartir d'une surface minimale dans S (qui est totalement g´eod´esiqueou qui correspond `aune solution de l'´equationde sinh-Gordon), une sous-vari´et´elagrangienne de dimension 3 de S3 × S3. Ici il faut remarquer que cette vari´et´elagrangienne n'est pas unique. En effet, pour chaque solution d'une ´equationdiff´erentielle suppl´ementaire il existe une telle vari´et´elagrangienne.
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