Duality in a special class of submanifolds and Frobenius manifolds

Dualit y in a s p ecial c l ass of submanifolds and F rob enius manifolds O.I. Mokhov Consider totally nonisotropic N - dimensional submanifolds M N in ( N + L )-di- mensional pseudo-Euclidean spaces E N + L k ( N -dimensional submanifolds that are not tangen t to is otropic cones of the am bien t ( N + L )-dimensional pseudo-Euclide an space at their p oints). Let ( z 1 , . . . , z N + L ) b e pseudo-Euclidean co o r dinates in E N + L k ; a submanifold M N is giv en b y a smo oth v ector-function r = ( z 1 ( u 1 , . . . , u N ) , . . . , z N + L ( u 1 , . . . , u N )), rank ( ∂ z i /∂ u j ) = N , 1 ≤ i ≤ N + L, 1 ≤ j ≤ N ; ∂ r/ ∂ u i = r i ( u ), 1 ≤ i ≤ N , is a basis of the tangent space at an a rbitrary p oint u = ( u 1 , . . . , u N ) of the sub- manifold M N ; n 1 ( u ) , . . . , n L ( u ) is an arbitrary basis of the normal space N u that dep ends smo othly on the p oint u ; g ij ( u ) = ( r i , r j ), 1 ≤ i, j ≤ N , is the first fun- damen tal form of the submanifold (( · , · ) is the pseudo-Euclidean scalar pro duc t in E N + L k ), h αβ ( u ) = ( n α , n β ), 1 ≤ α , β ≤ L (det g ij ( u ) 6 = 0 and det h αβ ( u ) 6 = 0 f o r totally nonisotropic submanifolds). The G auss and W eingarten decomp ositions hav e the for m ∂ 2 r ∂ u i ∂ u j = a k ij ( u ) ∂ r ∂ u k + b β ij ( u ) n β ( u ) , ∂ n α ∂ u j = c k αj ( u ) ∂ r ∂ u k + d β αj ( u ) n β ( u ) , (1) resp ectiv ely . The co efficien ts a k ij ( u ) , b β ij ( u ) , c k αj ( u ) , d β αj ( u ), the metric g ij ( u ) and the functions h αβ ( u ) satisfy a n um b er of relations including the Gauss equations, the Co dazzi equations and the Ricci equations for a n y submanifold. Note tha t in this pap er w e consider only the lo cal theory o f submanifolds. W e shall single out a sp ecial class of N -dimensional submanifolds in 2 N -dimensi- onal pseudo-Euclidean spaces. Our purp ose is to single out the case, when the sets of the basis v ectors of the tangen t and normal spaces, r i ( u ), 1 ≤ i ≤ N , and n α ( u ), 1 ≤ α ≤ L , p ossess equal righ ts and are dual one to another. It is obvious that the condition L = N is necessary for suc h dualit y . Moreo v er, another necessary condition is the p oten t ialit y of the basis n α ( u ): n α ( u ) = ∂ n/∂ u α , 1 ≤ α ≤ N , where n ( u ) is a certain ve ctor-function on the submanifold. Definition 1. A basis n α ( u ), 1 ≤ α ≤ N , in the normal space N u of a certain N -dimensional submanifold in a 2 N -dimensional pseudo-Euclidean space is called 1 The work was completed with the financial suppo rt of the Russian F oundation for Ba sic Resear ch (grant no. 08-0 1-000 54) and the Progra mme for Suppo rt of Leading Scientific Schools (gra nt no . NSh-1824.2 008.1). 1 p otential if there exists a vector-function (a p otential of the normals) n ( u ) on the submanifold suc h t ha t n α ( u ) = ∂ n/∂ u α , 1 ≤ α ≤ N . If the v ector- function n ( u ) exists, then it generates a p oten tial basis in the normal space N u in any co ordinate system. Definition 2. An N -dimensional submanifold in a 2 N -dimensional pseudo- Euclidean space is called a submanifold with p otential normal b asis (or a submanifold with p otential of normals ) if there exists a vec tor-function (a p otential of normals ) n ( u ) on the submanifold suc h that the v ectors n i ( u ) = ∂ n/∂ u i , 1 ≤ i ≤ N , form a basis in the normal space N u at any p oint u on the submanifold. Definition 2 is inv ariant (it do es not dep end on a lo cal co ordinate system). The co efficien ts a k ij ( u ) in the Gauss decomp osition (1) ar e alwa ys the co e fficien ts of the Levi-Civita connection of the metric g ij ( u ). If the submanifold is equipped with a p oten tial normal basis, then a similar assertion is also true for the co efficien ts d β αj ( u ) in the W eingarten decompo sition (1). Consider the corresp onding G auss and W eingarten decomp ositions: ∂ 2 r ∂ u i ∂ u j = a k ij ( u ) ∂ r ∂ u k + b k ij ( u ) ∂ n ∂ u k , ∂ 2 n ∂ u i ∂ u j = c k ij ( u ) ∂ r ∂ u k + d k ij ( u ) ∂ n ∂ u k . (2) Theorem 1. The functions h ij ( u ) = ( ∂ n/∂ u i , ∂ n/∂ u j ) de fi ne a c ovariant metric on the subm anifold, and the c o efficients d k ij ( u ) in the Weinga rten de c omp osition (2) ar e the c o efficients of the symm e tric affine c on n e ction c omp atible with the metric h ij ( u ) , that is the c o efficients o f the L evi-C i v i ta c on n e ction of the metric h ij ( u ) . The c o efficients b k ij ( u ) and c k ij ( u ) ar e tensors of typ e (1, 2) s ymm etric with r esp e ct to the subscripts i and j on the submanifold M N . Theorem 2 (dualit y principle). I f a subma nifold is e quipp e d with a p otential of normals n ( u ) and given by the v e ctor-functions ( r ( u ) , n ( u ) ) , then the ve ctor-functions ( n ( u ) , r ( u )) also give a submanif o ld e quipp e d with the p otential of norma ls r ( u ) such that ∂ n/∂ u i , 1 ≤ i ≤ N , ar e tangent ve ctors and ∂ r /∂ u i , 1 ≤ i ≤ N , ar e b asis normal ve ctors of the submanifold, and m or e over, in this c ase al l the o b je cts o f the lo c al the ory of such s ubm anifolds ar e dual to e ach other, in p articular, the Gauss de c om p osition b e c omes the Weingarten de c omp osition and the Weingarten de c omp osi tion b e c ome s the Gauss de c o mp osition, the Gauss e quations b e c omes the Ric ci e quations and the Ric ci e quations b e c omes the Gauss e quations, the Co dazzi e quations changes to themselves ( they ar e self-dual ) , the tensor b k ij ( u ) b e c om es the tensor c k ij ( u ) and the tensor c k ij ( u ) b e c omes the tenso r b k ij ( u ) , the metric g ij ( u ) b e c omes the metric h ij ( u ) and vic e versa. Submanifolds with p otential of normals form an imp orta n t and ric h class of sub- manifolds. Arbitrary one-dimensional submanifolds of pseudo-Euclidean planes are a trivial example of suc h su bmanifolds. The general theory o f submanifolds with p oten tial of normals, the dualit y principle for them and imp o r t an t examples will b e presen ted in a separate pap er. The submanifolds equipp ed with natural F ro b enius 2 structures, whic h w ere constructed by the presen t author in [1]–[5] and whic h realize arbitrary F rob enius manif o lds (the theory of F rob enius manifolds w as constructed in [6]), are a particular case of submanifolds with p ot ential of normals. The presen t author has pro ve d in [1] and [2] that an arbitrary F rob enius manifold can b e realized as a certain flat submanifold with p oten tial of normals for whic h g ij ( u ) = ch ij ( u ), c = const 6 = 0, where c is a deformation para meter preserving t he corresp onding F rob enius structure. Let h ij ( u ) = η ij , η ij = η j i , det η ij 6 = 0, η ij = const, g ij = cη ij , c = const 6 = 0. In this case, for the submanifolds with p o ten tia l of normals, the rela- tions a k ij ( u ) = d k ij ( u ) = 0 are satisfied, and also there exists a function Φ( u ) suc h that b k ij ( u ) = η k s ∂ 3 Φ /∂ u s ∂ u i ∂ u j , c k ij ( u ) = − (1 / c ) b k ij ( u ), η is η sj = δ i j , a nd all the relations o f the lo cal theory of submanifolds are satisfied if and only if the f unction Φ( u ) satisfies the asso ciativity equations of t w o -dimensional top ological quan tum field theories (the Witten–Dijkgraaf–V erlinde–V erlinde equations, see [6]) ∂ 3 Φ ∂ u i ∂ u j ∂ u s η sp ∂ 3 Φ ∂ u p ∂ u k ∂ u l = ∂ 3 Φ ∂ u i ∂ u k ∂ u s η sp ∂ 3 Φ ∂ u p ∂ u j ∂ u l . (3) Bibliograph y [1] O.I. Mokho v, F rob e nius ma nif o lds as a sp ecial class of submanifolds in pseudo- Euclidean spaces, will b e published in AMS T ranslations Series 2, V olume 224, Ad- v ances in Mathematical Sciences, eds. V.M.Buc hstab er and I.M.Krich ev er, “Geome- try , T op olo gy , and Mathematical Ph ysics: S.P .No vik ov’s Seminar: 2006–20 07”, Amer. Math. So c., Pro vidence, RI 2008; h tt p:/ /arXiv.org/abs/0710 .5860 (2007 ). [2] O.I. Mokhov, Theory of submanifolds, the asso ciativity equations in 2D to p o- logical quan tum field theories, and F rob e nius manifolds, In: Pro ceedings of t he W orkshop “Nonlinear Phys ics. Theory and Ex p erimen t. IV.” (Gallip oli, Lecce, Italy , 22 June – 1 July , 2006), published in T eoret. i Matemat. Fizik a 152 :2 (2007), 368 –376; English transl., Theoret. Math. Ph ys. 152 :2 (2007), 1183– 1190; Preprin t MPI 06 -152, Max-Planc k- Institut f ¨ ur Mathematik, Bonn, G ermany , 2006; arXiv:math.DG/061 0933 (2006 ) . [3] O.I. Mokho v, Submanifolds in pseudo-Euclidean spaces and D ub o vin–F rob e nius structures, In: Pro c eedings of the 10th International Confrence “Differential G eome- try and its Applications” in honour of the 3 00th anniv ersary o f the birth o f Leonhard Euler, August 27–3 1 , 2007 , O lo mouc, Czec h Republic, W orld Scien tific, Singap ore, 2008, pp. 505 –516. [4] O.I. Mokhov , Non-lo cal Hamiltonian op erators of h ydro dy namic ty p e with flat metrics, and the asso ciativity equations, Usp ekhi Mat. Nauk 59 :1 (2004) , 187–188; English transl., Russian Math. Surv eys 59 :1 (20 0 4), 191–192 . 3 [5] O .I. Mokho v, Nonlo cal Hamiltonian op erators of h ydro dy namic t yp e with flat metrics, inte grable hierarc hies, and the asso ciativit y equations, F unktsional. Anal. i Prilozhen. 40 :1 (2 006), 14–29; English transl., F unct. Anal. Appl. 40 :1 (200 6), 11–23; arXiv:hep-th/040 6292 ( 2 004). [6] B. Dubrovin, Geometry of 2D top ological field theories, In: Lecture Notes in Math., vol. 162 0, Springer-V erlag, Berlin, 1996, pp. 120–348; a r Xiv:hep-th/9 4 07018 (1994). O.I. Mokhov Cen tre fo r Non-Linear Studies, Landau Institute for Theoretical Ph ysics, Russian Academ y of Sciences; Departmen t o f Geometry and T op ology , F acult y o f Mec hanics and Mathematics, Lomonoso v Moscow State Univ ersity E-mail : mokho v@mi.ras.ru; mokhov @landau.ac.ru; mokhov@bk.ru 4