Harmonic maps on contact metric manifolds

In this paper, we study some harmonic or 03C6-pluriharmonic maps on contact metric manifolds.


Introduction
The theory of harmonic maps on Riemannian and Kahler manifolds has been developed in the last thirty years by many authors (see [E-L], [E-R], [LI], [M-R-S] and their references). In odd dimension, the almost contact metric manifolds represent the analogue of almost hermitian manifolds (see [BL]).The first geometers to consider almost contact manifolds where W.Boothby -H.Wang [B-W], J.Gray [GR] and P.Libermann [LB]. A systematic study of them with adapted Riemannian metrics was initiated Sasaki and Ills school.
A reference book for this subject is the one of D.E.Blair ([BL2]). In the seventies there have been introduced interesting generalizations of almost contact metric structures by Blair S.Goldberg and K.Yano [G-Y], R.Lutz [LU]. There exists now v rich literature concerning the theory of harmonic maps in Kahler manifolds and more generally in almost hermitian manifolds. The purpose of this paper is tu initiate the study of harmonic maps into almost contact metric manifolds. Moreover we introduce the concept of 03C6-pluriharmonicity in analogy with the known one from the geometry of almost hermitiau manifolds.
In section 2 we consider 03C6-holomorphic maps between two contact metric manifolds and we prove that they are harmonic maps. The third section is devoted to the study of 03C6-pluriharmonicity on almost contact metric manifolds. If f : ~ M' is a 03C6-holomorphic map. between Sasaki manifolds, then f is 03C6-pluriharmonic if and only if it is an isometric immersion. In the fourth section we study the harmonicity and 03C6-pluriharmonicity on globally 03C6-symmetric Sasaki manifolds. Finally, in section 5 we give some examples. Acnowledgment The authors thank Prof.D.E.Blair who made them attentive on Olszak's paper. and suggested a simple proof of Theorem 2.2. The first author wishes to thank Prof.L.Lemaire for several discussions on the theory of harmonic maps, which took place during his staying in Bruxelles in the spring 1992. The second author was partially supported by MURST 40 %.

2
Harmonic maps on contact metric manifolds Let (M, g) and (VI', g') be two Riemannian manifolds and f : ~ VI' a differentiable map. All manifolds and maps are supposed to be of class C~. We denote by V, V' the Levi-Civita connection on M and M' respectively and by V the connection induced by the map f on the bundle f-1(TM'). Then the second fundamental form a f of f is defined as follows: (1) _for any E X(M), where X(M) denotes the space of differentiable vector fields on M.
We often will write a and f * instead of 03B1f and df respectively. is an orthonormal basis for the tangent space TxM at x E M. We say that a map f : lyI --~ between Riellanliau manifolds aI and is a harmonic map iff r( f = 0.
Let ~l be a differentiable manifold of dimension Z~a + 1.
Recall that an almost contact structure on NI is a triple (03C6, 03BE, y), where 03C6 is a tensor field of type (1,1), ~ is a vector field and 1J is a 1-form which satisfy: (3) = -I + r) ~ ~ and zl(~) =1 where I is the identity endomorphism on TM. These conditions imply (4) 03C603BE = 0 and ~ o 03C6 = 0 Furthemore, if g is all associated Riemannian n ietric ou M , that is a metric that. satisfies (5) 03C6Y) = g(X, Y) -~(X)~(Y) X,Y E X(A/) ) then we say that (03C6, 03BE, rj, g) is all almost contact metric structure. In such a way we obtain an almost contact vaetric manifold which is the analougue of an almost herniitian manifold . We denote by ~ the fundamental 2-form on M defined by ~(,~', ~' ) = g(.~'. ~,,I~~~. An almost contact metric structure is called a contact nlf.tric structure iff 03A6 = d~. An odd-dimensional manifold is called a contact manifold if it carries a contact metric structure. In [MA] .J.Martinet proved that every compact orientable 3-manifoldI carries a contact structure. An almost contact structure (03C6, 03BE, ~) is normal if where is the Nijenhuis torsion tensor of ~?. yVe denote by D the distribution orthogonal to 03BE and by r(D) the space of differeutiable sections of D. Using the Levi-Civita connection determined by g we can define a Sasaki manifold as an almost coiitact metric manifold such that for any X, Y E T(:LI), (see (BL~).
REMARK In the hypotheses of the previous proposition we have dim M'.
If we put y = for any X E r(D), we obtain from (2.7), tl ~ + , p 1" w 0 , 4K , 1" e ,I ( lil ) where 03B1 is the second fundamental form of f . Furthermore, f is said to bf Dpluriharmonic if (3.1) holds for any X,Y E r(D) , Obviously. 03C6 -pluriharmonicity implies D -pluriharmonicity.
Proposition 3.2 Let f : lyI -+ be a 03C6-pluriharmonic map from arr almost contact metric manifold into a Riemannian manifold. Then f ia a harmonic map.
Proof. Fixed a local orthonormal p-basis (e1,...,en; 03C6(e1),...03C6(en),03BE) in M. m I=I so that f is harmonic ©. Let TcM be the complexification of Then 03C6 can be uniquely extended to a complex linear endomorphism of TcM, denoted also by 03C6 which satisfies (2.4). The eigenvalues of 03C6 are therefore i, 0, and -i. We consider the usual decomposit ion TC M = T+M ~ TO M c~ T'M of TcM in the eigenbundles corresponding to the eigenvalues I, 0, -I of ys. Proposition 3.3 Let be an almost contact metric .manifold with the structure (w, j, .q, g) and M' a Riemannian manifold. Then, for a map f : lkI -M' uir have that f is 03C6-pluriharmonic if and only if the. following conditions hold: I) a(Z, "'l'°') = 0 for any Z e r(T+ ikf) it) a(.K, () = 0 for any ,I e F(D) i ii) () = 0 Proof If Z e t(T+ we have Z = X -i03C6X with X e F( D ) and Z = ," + i03C6X, so that £t( Z, 2) * 03B1(X, X) + 03C6X) * 0 by the 03C6-pluriharmonicity of f. Using (2.4) we obtain it) and iii).
It is well known that a Riemannian submersion is a harmonic map if and only if its fibres are minimal submanifolds ( [ER]). If rra = n and M is a Sasaki manifold, then 03BE is a geodesic vector field, so that 03C0 is a harmonic map. It is easy to proof the following proposition. for any X E ) c) 03BE is a Killing vector f ield and d~ = o. Remark If tyl is a contact metric manifold, d~ ~ 0 and 03C0 can not be 03C6-pluriharmonic. Proposition 4.2 Let NI ~ B be a 03C6-holomorphic Riemannian submersion, where M is a (2n+1)-dimensional almost contact metric manifold and B is a 'rrdimensional almo.st hermitian manifold. If NI' is a Riemannian manifold and f : Proof Since the condition f*03BE = 0 implies that f is constant on the fibres. it follows that f is uniquely determined.
Then we know that lbI is a principal bundle over a 21i-dilnensional hermitian globally symmetric space B with a Lie group G of dimension 1, isomorphic to the 1-parameter group of global transformation spanned by 03BE. The projection M ~ B is a Riemannian submersion and the connection 1-form on is deterniined by rt. Furthermore. 03C0 is a 03C6-holomorphic and harllollic map, since ç is a geodesic vector field.
In this section we suppose that M(c) is a (27a + 1-dimensional globally 03C6-symmetric Sasaki manifold with constant p-sectional curvature c. Then, we know that A/(c) is a principal bundle over a globally symmetric hermitian space of constant holomorphic sectional curvature c'. Furthermore, if we suppose c > 0 then B has to br bi-holonorphic to the complex projective space Cf with the Fubini-Study metric (since c' > 0.) Proof Since f*~ = 0, the previous proposition implies that f is uniquely determined.
Denote by .J and J' the complex structures on (.'P" and respectively. Suppose that f is ±03C6 -holomorphic. For any X', Y' E ,t' (CP"), let X,Y E ,'t' (M) be such that X' = 03C0*X, Y' = 03C0*Y. Since ~r is ..p -holomorphic, we have: If f is a D-pluriharmonic map, then 7 is pluriharlouic. It is well known that a pluriharmonic map 1 : CP" ~ CPm is ± holomorphic map (see, for example [OH] is said to be regular if every point J-E AI has L cubical coordinate neighborhood U such that the integral curves of ( passing through f' pass trough the neighborhood only once. If (03C6, 03BE, ~) is an almost contact structure with F regular on a compact manifold then M is a principal circle bundle over a manifold B (the set of maximal integral curves with the quotient topology) and we denote the projection by 7r. Furthermore, ~ is a connection form on M. Let g be a adapted metric on M. yVe define an almost complex structure .7 and a Riemannian metric lu on B as follows: ( 1) h(X, Y) = g(X*, Y*) o 03C0 X, Y E ) (2) .L~' ~ ~*(~,~'*) where .~'*, Y* denote lifts of .x and Y' respectively with respect to the connection y (see [B-W], [OG].
It is easy to see that 03C0 is a Riemannian submersion satisfying 03C0*03C6 = J03C0*.
If Af is Sasaki, then 03C0 is a harmonic map.
A particulary well-known example of this fibration is the S2n+1 circle bundle over CPn.
Odd-dimensional Lie groups and, nlore generally, odd-dimensional parallelizable manifolds admit almost contact metric structure (see [BL2]). Now we generalize tllr above examples. Let N(J, h) be an almost hermitian manifold and denote by CT a Lie group with left-invariant metric , >. Let 03BE1,...,03BE3 be an orthonormal basis of the Lie algebra g and (~1, ..., the dual 1-forms. We denote by P a principal bundle on 1V with projection 7r and connection 1-form 03C9 = 03A33i=1 + i ~ 03BEi which takes valurs ill g. For any X (N) let XH be its horizontal lift and denote by A* the fundamental vertical vector field corresponding to A E g. We define a tensor field 03C6 of type (1,!) on P. putting (3) = (JX)H and 03C6(A*) = 0 and we consider a nietric g on P defined by (4) 7T~ 0 (~ = By a straightforward computation we obtain 7'(7r) = 0, so that 7r is a harmonic map. Furthermore 03C0 is 03C6-pluriharmonic if and only if 03C9 is flat.