Habilitation Thesis DIFFEOMORPHISM GROUPS AND GEOMETRIC MECHANICS CORNELIA VIZMAN

Habilitation Thesis DIFFEOMORPHISM GROUPS AND GEOMETRIC MECHANICS CORNELIA VIZMAN

Chapter 1 Summary 1.1 Rezumat Numitorul comun al activităţilor mele de cercetare este geometria diferenţială infinit dimensională, fie ca obiect de studiu, fie ca instrument de lucru. Obiecte matematice infinit dimensionale importante şi des întâlnite sunt grupurile de difeomorfisme. M-am ocupat de acestea începând cu teza de doctorat despre orbite coadjuncte ale unor grupuri infinit dimensionale, în special grupuri de difeomorfisme, dar mai târziu grupurile de difeomorfisme şi algebrele Lie de câmpuri de vectori m-au adus tot mai aproape de mecanica geometrică, în special dinamica fluidelor. Odată cu articolul fondator al lui Arnold [5], formalizat apoi de Ebin şi Marsden [13], studiul ecuaţiilor lui Euler pentru fluidul ideal a fost transferat către studiul geodezicelor unei metrici Riemanniene drept invariante pe grupul difeomorfismelor ce păstrează volumul. Aceasta a fost prima cărămidă a hidrodinamicii geometrice (programul lui Arnold). Numeroase alte sisteme de tip hidrodinamic se pot scrie ca ecuaţii geodezice (sau, mai general, ca ecuaţii Euler-Poincaré) pe grupuri de difeomorfisme sau pe extensii ale acestora, astfel putând fi folosite metode geometrice pentru studiul lor. În consecinţă rezultatele mele obţinute după susţinerea doctoratului se pot încadra în două mari direcţii: 1. Grupuri infinit dimensionale, extensii şi orbite coadjuncte: [2], [22], [61], [62], [74], [76], [77], [78], [80], [84], [85]; 2. Hidrodinamică geometrică, ecuaţii Euler-Poincaré şi geometria lor: [17], [18], [19], [20], [26], [72], [75], [81], [82], [83], [86], [87], [88], [89], [90]. 1. Grupuri infinit dimensionale, extensii şi orbite coadjuncte. Un rezultat cu aplicaţii surprinzătoare în teoria cuantică a gravitaţiei se referă la Grassmannienii neliniari, care sunt spaţii de subvarietăţi, eventual cu anumite proprietăţi suplimentare, ale unei varietăţi diferenţiabile fixate. Împreună cu 7

8 Haller am arătat în [22] că Grassmannienii neliniari de subvarietăţi de codimensiune doi sunt orbite coadjuncte ale extensiilor centrale ale grupului difeomorfismelor ce păstrează volumul. Şi printre orbitele coadjuncte ale grupului difeomorfismelor Hamiltoniene regăsim Grassmannieni neliniari, anume Grassmannienii neliniari de subvarietăţi simplectice. Binecunoscuta extensie centrală de precuantificare, datorată lui Kostant [42] şi Souriau [71], constă în grupul cuantomorfismelor ca extensie centrală a grupului difeomorfismelor Hamiltoniene ale unei varietăţi simplectice precuantizabile. Un rezultat obţinut împreună cu Neeb [61] generalizează această extensie centrală pentru cazul unei varietăţi infinit dimensionale. Recent am descoperit o versiune mai generală a extensiei centrale de precuantificare, anume o extensie abeliană asociată unei 2-forme diferenţiale închise cu valori vectoriale [76]. Toate aceste extensii de precuantificare pot fi folosite pentru construcţia unor extensii de grupuri Lie infinit dimensionale prin procedeul de pull-back. Alt mod de construcţie, folosind grupul drumurilor, l-am prezentat în lucrarea [80]. Alte rezultate care se referă la extensii de grupuri Lie au un mai pronunţat caracter algebric. În articolul [2] am studiat împreună cu Alekseev şi Severa extensii ale algebrelor Lie de curenţi ce se pot obţine cu ajutorul a doi functori definiţi în [1], în special simetriile modelului sigma. Aceşti functori asociază o algebră Lie unei perechi ce constă dintr-o varietate diferenţiabilă şi o algebră Lie diferenţială graduată. În alt articol [62], scris în colaborare cu Neeb, am propus o interpretare abstractă pentru acţiunea Hamiltoniană a unui grup şi extensiile centrale de grupuri Lie asociate în cazul general când forma simplectică se înlocuieşte cu un 2-cociclu de algebră Lie. În ultimul meu articol [74] m-am ocupat cu grupul k-jeturilor J k G al unui grup Lie G: descriind explicit operaţia de grup în trivializarea la dreapta, am determinat cociclul de grup pentru grupul J k G descris ca extensie abeliană a grupului J k 1 G. 2. Hidrodinamică geometrică, ecuaţii Euler-Poincaré şi geometria lor. Unul dintre cele mai bune rezultate ale mele a fost obţinut împreună cu Gay-Balmaz [19] şi se referă la perechea duală pentru ecuaţia lui Euler: am lămurit complet problema perechii duale de aplicaţii moment asociate, în spiritul articolului lui Marsden şi Weinstein [51] unde această pereche duală a apărut în contextul variabilelor Clebsch. A fost necesară introducerea extensiilor centrale ale unor grupuri de difeomorfisme (ce păstrează forma simplectică, respectiv forma volum) şi demonstrarea tranzitivităţii acţiuni acestor grupuri pe varietatea Fréchet a scufundărilor. Pentru manipularea formelor diferenţiale pe spaţiile de funcţii folosite aici, ne-am folosit de aşa numitul hat calculus, construit în acest scop în [75]. Acelaşi program l-am dus la bun sfârşit şi pentru ecuaţia Euler-Poincaré pe grupul difeomorfismelor (ecuaţia EPDiff) ce generalizează ecuaţia Camassa-Holm n-dimensională. Perechea duală EPDiff a fost introdusă de către Holm şi Marsden în [24], piciorul stâng fiind răspunzător pentru soluţii singulare, iar cel drept pentru constante de

9 mişcare. Perechile duale pentru ecuaţii Euler-Poincaré pe grupuri de automorfisme ale unui fibrat principal (ecuaţiile EPAut şi EPAut incompresibilă) le-am determinat împreună cu Gay-Balmaz şi Tronci în [17]. A fost necesară definirea unui nou grup de difeomorfisme, grupul cromomorfismelor, ce integrează o extensie centrală a algebrei Lie de câmpuri vectoriale Hamiltoniene ce admit funcţii Hamiltoniene invariante. Tranzitivitatea acţiunilor grupurilor de difeomorfisme implicate am demonstrat-o deocamdată în cazul unui fibrat principal trivial în articolul [18]. O familie de perechi duale asociate cu fenomenul de rezonanţă e prezentată într-un articol scris în colaborare cu Holm [26]. Sunt exemple de perechi duale asociate unor sisteme Hamiltoniene superintegrabile. Cuantificarea fluxului magnetic poate fi privită ca o condiţie de integrabilitate pentru extensia centrală Lichnerowicz a algebrei Lie a câmpurilor de vectori având divergenţa zero, am arătat în articolul [89]. În acest caz atât ecuaţiile de mişcare ale fluidului ideal încărcat electric, cât şi ecuaţia de superconductivitate, ambele într-un câmp magnetic dat, sunt ecuaţii geodezice pe grupul automorfismelor unui fibrat principal ce păstrează volumul, dotat cu metrică Riemanniană drept invariantă. Articolul[20] se ocupă cu subgrupuri total geodezice ale unui grup Lie dotat cu metrică Riemanniană drept invariantă. Pentru metrica L 2 drept invariantă am determinat toate varietăţile Riemanniene ce au proprietatea că subgrupul difeomorfismelor exacte ce invariază forma volum este total geodezic în grupul difeomorfismelor ce păstrează forma volum. Acestea sunt produsele răsucite dintre un tor plat şi o varietate simplu conexă. Astfel că am putut răspunde la întrebarea: în ce condiţii fluidul ideal păstrează proprietatea de a admite un potenţial. În colaborare cu Tığlay am studiat în [72] atât ecuaţii Euler-Poincaré generalizate, cât şi ecuaţii Euler-Poincaré pe spaţii omogene. Cu ecuaţii geodezice pe grupuri de difeomorfisme dotate cu metrici Riemanniene drept invariante ce au relevanţă în fizică şi geometria lor m-am ocupat şi în articolele [81], [82], [86], [87], [88] şi [90]. De asemenea am scris un articol tip survey [83] şi o monografie ştiinţifică [79] pe această temă. 1.2 Résumé The common denominator of my research is differential geometry in infinite dimensions, either as object of study, or as instrument of work. Important and widely spread infinite dimensional objects are diffeomorphism groups. I worked on them starting with my doctoral thesis on coadjoint orbits of infinite dimensional Lie groups, especially diffeomorphism groups, but later diffeomorphism groups and their Lie algebras of vector fields brought me closer to geometric mechanics, especially fluid dynamics.

10 Through the founding article of Arnold [5], later formalized by Ebin and Marsden [13], the study of Euler s equations for ideal fluids has been transfered to the study of geodesics on the group of volume preserving diffeomorphisms with right invariant Riemannian metric. This was the first brick of geometric hydrodynamic (Arnold s program). Many other systems of hydrodynamical type can be written as geodesic equations (or, more generally, as Euler-Poincaré equations) on diffeomorphism groups or on extensions of diffeomorphisms groups, thus making possible the use of geometric methods for their study. The results obtained after my PhD could be split in two big directions: 1. Infinite dimensional groups, extensions and coadjoint orbits: [2], [22], [61], [62], [74], [76], [77], [78], [80], [84], [85]; 2. Geometric hydrodynamics, Euler-Poincaré equations and their geometry: [17], [18], [19], [20], [26], [72], [75], [81], [82], [83], [86], [87], [88], [89], [90]. 1. Infinite dimensional groups, extensions and coadjoint orbits. A result with surprising applications in quantum gravity refers to nonlinear Grassmannians: spaces of submanifolds of a given differential manifold, sometimes with additional properties. In [22], together with Haller, we showed that the nonlinear Grassmannians of submanifolds of codimension two are coadjoint orbits of central extensions of the group of volume preserving diffeomorphisms. Also among the coadjoint orbits of the group of Hamiltonian diffeomorphisms one finds nonlinear Grassmannians, namely the nonlinear Grassmannians of symplectic submanifolds. The well known prequantization central extension, due to Kostant [42] şi Souriau [71], expresses the group of quantomorphisms as a central extension of the group of Hamiltonian diffeomorphisms of a prequantizable symplectic manifold. A result obtained together with Neeb [61] generalizes this central extension to the case of an infinite dimensional manifold. Recently a more general version of the prequantization central extension came up, namely an abelian extension associated to a closed vector valued differential 2-form [76]. All these prequantization extensions can be used to build extensions of infinite dimensional Lie groups by pull-back. Another method, that uses the path group of the infinite dimensional Lie group, is presented in [80]. Other results that refer to Lie group extensions have a more algebraic flavor. In [2], together with Alekseev and Severa, we studied extensions of current Lie algebras obtained using two functors introduced in [1], called current algebra functors. They associate a Lie algebra to a pair consisting of a differential manifold and a differential graded Lie algebra. One of the extensions we obtain is related to the sigma model symmetries. In collaboration with Neeb [62] we propose an abstract interpretation for Hamiltonian group

11 actions and associated central Lie group extensions in the general case when the symplectic form is replaced by a Lie algebra 2-cocycle. My last paper [74] deals with the group J k G of k-jets of a Lie group G: it describes explicitly the group multiplication in the right trivialization and determines the group 2-cocycle for J k G as an abelian extension of J k 1 G. 2. Geometric hydrodynamics, Euler-Poincaré equations and their geometry. One of my best results, obtained together with Gay-Balmaz [19], deals with the dual pair associated to Euler s equation. We completely solved the problem of the dual pair of momentum maps in the spirit of the paper [51] due to Marsden and Weinstein, where it first appeared in the context of Clebsch variables. It required to use central extensions of groups of diffeomorphisms (those preserving the symplectic resp. the volume form) and to prove the transitivity of their actions on the Fréchet manifold of embeddings. To manipulate differential forms on space of functions, we introduced the so called hat calculus [75]. The same program was completed also for the Euler- Poincaré equation on the diffeomorphism group (EPDiff) that generalizes the n-dimensional Camassa-Holm equation. The EPDiff dual pair was introduced by Holm and Marsden in [24]: the left leg is responsible for singular solutions, while the right leg provides constants of motion. Dual pairs for Euler-Poincaré equations on the automorphism groups of a principal bundle (EPAut and incompressible EPAut) were determined together with Gay-Balmaz and Tronci in [17]. We needed to introduce a new diffeomorphism group, the chromomorphism group, that integrates a central extension of the Lie algebra of Hamiltonian vector fields that admit invariant Hamiltonian functions. The transitivity of the actions of the corresponding diffeomorphism groups was proved for the moment in the special case of trivial principal bundles [18]. A family of dual pairs associated with resonances is described in a joint paper with Holm [26]. These are examples of dual pairs related to superintegrable Hamiltonian systems. That quantification of magnetic flux can be seen as an integrability condition for the Lichnerowicz central extension of the Lie algebra of divergence free vector fields is shown in [89]. In this case both the charged ideal fluid motion and the superconductivity equation in a fixed magnetic field can be expressed as geodesic equations on the group of volume preserving automorphisms of a principal bundle, endowed with right invariant Riemannian metric. In the article [20] we consider totally geodesic subgroups of a Lie group endowed with right invariant Riemannian metric. For the L 2 metric we determine all Riemannian manifolds that satisfy the following property: the subgroup of exact volume preserving diffeomorphisms is totally geodesic in the group of volume preserving diffeomorphisms. These are twisted products of flat tori with simply connected manifolds. Thus we answered the question: when does the ideal fluid flow preserve the property of possessing a potential.

12 In a joint work with Tığlay [72] we study generalized Euler-Poincaré equations, as well as Euler-Poincaré equations on homogeneous spaces. Several geodesic equations on diffeomorphism groups endowed with right invariant Riemannian metrics with physical relevance and some of their geometric properties can be found in [81], [82], [86], [87], [88] and [90]. I have also written a survey article [83] and a scientific monograph [79] on this subject.

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