The Erwin Schrodinger International Boltzmanngasse ESI Institute for Mathematical Physics A Wien Austria Hermitian and Kahler Submanifolds of a Quaternionic Kahler Manifold DV Alekseevsky S Marchiafava Vienna Preprint ESI January Supp orted by Federal Ministry of Science and Transp ort Austria Available via httpwwwesiacat HERMITIAN AND KAHLER SUBMANIFOLDS OF A QUATERNIONIC KAHLER MANIFOLD DV ALEKSEEVSKY S MARCHIAFAVA Sophus Lie Center Moscow Universita di Roma I Abstract We study almost Hermitian submanifolds of a quaternionic Kahler mani 4n fold M Q g We investigate when such submanifold is Hermitian almost Kahler and Kahler Some characterizations of Kahler submanifolds are given In the second part of the pap er we classify Kahler manifolds which can b e realized as Kahler sub manifolds of a quaternionic Kahler manifold with parallel non zero second fundamental form These manifolds can b e characterized as lo cally Hermitian symmetric spaces which admit a parallel line bundle of cubic forms and coincide with Tsukada manifolds which can b e realized as Kahler submanifolds of the quaternionic pro jective space with parallel non zero second fundamental form Introduction n f Let M Q ge b e a quaternionic Kahler manifold where ge is the metric and n Q EndTM is the quaternionic structure We will always assume that M has m n f non zero scalar curvature A submanifold M M with induced metric g is called an almost Hermitian submanifold if it is given a g orthogonal almost complex structure J on M which is induced from a section J of the bundle Q M This jM means that J T M T M x M and J J x x jTM m An almost Hermitian submanifold M J g is called Hermitian if the almost complex structure J is integrable almost Kahler if the Kahler form F g J is closed and Kahler if F is parallel In the rst section we study an almost Hermitian m n f submanifold M J g of a quaternionic Kahler manifold M We give dierent conditions for an almost Hermitian submanifold to b e Hermitian For example we prove that any analytic complete almost Hermitian submanifold M of the quater n f nionic Kahler manifold M Q ge with p ositive scalar curvature is Hermitian if m dim M k Theorem We prove that any almost Kahler submanifold M R Mathematics Subject Classication C C Key words and phrases Quaternion Kahler manifold almost Hermitian submanifold Kahler submanifold Parallel submanifold Work done under the program of GNSAGA of CNR partially supp orted by MURST Italy and ESI Vienna DV ALEKSEEVSKY S MARCHIAFAVA n f of a quaternionic Kahler manifold M is Kahler and hence a minimal submanifold and give some lo cal characterizations of such submanifold In particular by complet ing a known result of K Tsukada we prove that an almost Hermitian submanifold M is Kahler if and only if it is total ly complex ie it satises the condition J T M T M x M x x where J is a section of Q M which anticommutes with J In section jM n n f we study Kahler submanifolds M in a quaternionic Kahler manifold M Q ge Using the isomorphism J TM T M b etween the tangent and the normal bundle we identify the second fundamental form h of M with a tensor C J h TM S T M This tensor at any p oint x M b elongs to the rst prolongation of the space S End T M of symmetric endomorphisms anticommuting with J and J x the asso ciated covariant tensor g C has the form g C q q where q S T M is x a holomorphic cubic form The GaussCo dazzi equations written in terms of the tensor C take a simple form We show that the second GaussCo dazzi equation is equivalent to the rst In subsec n tion we study the case when M is a Kahler submanifold of a lo cally symmetric n f quaternionic Kahler space M We get the necessary and sucient conditions for n M to b e a lo cally symmetric manifold in terms of the tensor C In particular if e f M is curvature invariant ie if the curvature R of M at a p oint x M satises the condition e R T T T T T T M x then M is a lo cally Hermitian symmetric manifold if and only if the unvalued form C C X Y C C X Y TM X Y which satises the rst and the second Bianchi identity is parallel n f If M is a quaternionic space form then any Kahler submanifold M is curvature invariant Hence it is symmetric if and only if the tensor C C is parallel n The section is devoted to a classication of Kahler submanifolds M of a quater n f nionic Kahler manifold M with parallel non zero second fundamental form h or shortly paral lel Kahler submanifolds In terms of the tensor C this means that r C X J C X TM X where is the form dened by and r is the LeviCivita connection of M We prove that any parallel submanifold M J g which is not totally geo desic ad mits a parallel holomorphic line subbundle L of the bundle S T M such that the L connection induced on L has the curvature R i g J where is the reduced n n f scalar curvature of M We give the classication of all such Kahler manifolds M with parallel holomorphic line bundle of cubic form All of them are Hermitian sym metric spaces Moreover the remarkable Tsukada results Tsu shows that all these HERMITIAN AND KAHLER SUBMANIFOLDS OF A QUATERNIONIC n manifolds M admits an explicitly describ ed realization as a non totally geo desic n parallel Kahler submanifold of the quaternionic pro jective space H P The similar n problem of realization of M as parallel Kahler submanifolds of other Wolf spaces ie symmetric quaternionic Kahler spaces remains op en Almost Hermitian submanifolds of a n f quaternionic Kahler manifold M Q ge n f The structure equations of a quaternionic Kahler manifold M n Let M Q g b e a quaternionic Kahler manifold that is a Riemannian manifold n M g of dimension n with parallel quaternionic structure Q ie a rank sub bundle of the bundle of endomorphisms lo cally spanned by a triple of lo cally dened anticommuting g orthogonal almost complex structures H J J J J J H is called a local basis of Q Since Q is parallel one can write e rJ J J e where r is the LeviCivita connection the are lo cally dened forms and is a cyclic p ermutation of Moreover if n then the metric g is Einstein Be and the forms satisfy the following structure equations see AM Th d F where is the reduced scalar curvature related to the scalar curvature K by K nn F g J are Kahler forms and the exterior dierential f of a form is given by d X Y X Y Y X X Y X Y T M By taking the exterior derivative of we get dF F F e We recall also that the following identities for the curvature tensor R hold e RX Y J F X Y J F X Y J For n the formula and all the following results remain true if we assume that the metric g is Einstein and antiselfdual ie the selfdual part W of the Weyl tensor vanishes This will b e assumed in the following n f Almost Hermitian submanifolds of M m n f Let M g b e a submanifold of a quaternionic Kahler manifold M Q ge with induced metric g eg and J is a g orthogonal almost complex structure on jM DV ALEKSEEVSKY S MARCHIAFAVA m m f M The manifold M J g is called an almost Hermitian submanifold of M if there is a section J M Q such that jM J T M T M x M x x and J J jTM If the complex structure J is integrable then M J g is called an Hermitian submanifold Remark Note that the section J of Q is uniquely dened by J jM m For any p oint x M we can always cho ose a lo cal basis H J J J J J n f e of Q dened in a neighb orho o d U of x in M such that J J We will jT M U m call it an adapted basis for M J g Since our considerations are lo cal we will m assume for simplicity that U M and we put F F jM jM T M the maximal quaternionic ie Qinvariant subspace For any x M we denote x of the tangent space T M and write x T M T M D x x x where D is the orthogonal complement Note that if J J J is an adapted basis x T M T M J T M in a p oint x M then x x x f f Recall that if M is a submanifold of a Riemannian manifold M ge and T M x f M at p oint M is the orthogonal decomp osition of the tangent space T T M T x x x e x M then the covariant derivative r in the direction of a vector X T M can X x b e written as r A X X e r X t r A X X that is e e r Y r Y hX Y r A X r X X X X for any tangent vector eld Y and any normal vector eld on M Here r is X the covariant derivative of the induced metric g on M r is the normal covariant X derivative in the normal bundle T M which preserves the normal metric g t gjT M A Y hX Y T M is the second fundamental form and A A X X X where A End TM is the shape operator asso ciated with a normal vector We will use this notation in the sequel m Theorem AM Let M J g m be an almost Hermitian submani n f Q ge Then fold of the quaternionic Kahler manifold M the almost complex structur J is integrable if and only if the local form m J on M associated with an adapted basis H J vanishes HERMITIAN AND KAHLER SUBMANIFOLDS OF A QUATERNIONIC J is integrable if one of the fol lowing condition hold a dimD on an open dense set U M x b M J is analytic and dimD at some point x M x c D and J D D x M x x x d M J is analytic and J D D on a nonempty open set U M x x Proof Remark that if M is an almost complex submanifold of an almost complex manifold N J then the restriction of the Nijenhuis tensor N to the submanifold J 1 M coincides with the Nijenhuis tensor N of the almost complex structure