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The Structure of Harmonic Morphisms with Totally Available at: http://www.ictp.trieste.it/~pub off IC/99/90 United Nations Educational Scienti c and Cultural Organization and International Atomic Energy Agency THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS THE STRUCTURE OF HARMONIC MORPHISMS WITH TOTALLY GEODESIC FIBRES 1 M.T. Mustafa Faculty of Engineering Sciences, GIK Institute of Enginneering Sciences and Technology, Topi-23460, N.W.F.P., Pakistan and The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. Abstract The structure of lo cal and global harmonic morphisms b etween Riemannian manifolds, with totally geo desic bres, is investigated. It is shown that non-p ositive curvature obstructs the existence of global harmonic morphisms with totally geo desic bres and the only such maps from compact Riemannian manifolds of non-p ositive curvature are, up to a homothety, totally geo desic Riemannian submersions. Similar results are obtained for lo cal harmonic morphisms with totally geo desic bres from op en subsets of non-negatively curved compact and non-compact manifolds. During the course, of this work we prove the non-existence of submersive harmonic morphisms with totally geo desic bres from some imp ortant domains, for instance from compact lo cally symmetric spaces of non-compact typ e and op en subsets of symmetric spaces of compact typ e. MIRAMARE { TRIESTE August 1999 1 Regular Asso ciate of the Ab dus Salam ICTP. E-mail: [email protected] 2 1. Introduction Harmonic morphisms are maps b etween Riemannian manifolds which preserve germs of har- monic functions, i.e. these lo cally pull back real-valued harmonic functions to real-valued har- monic functions. These are characterized as a sub class of harmonic maps, precisely, these are harmonic maps which are horizontally weakly conformal. What is sp ecial ab out this character- ization is that it endows harmonic morphisms with analytic as well as geometric prop erties. On the other hand, it puts strong restrictions on their existence as solutions of an over-determined system of partial di erential equations. The purp ose of this article is to study questions re- lated to the existence and structure of harmonic morphisms, with totally geo desic bres, from compact and non-compact Riemannian manifolds. The Bo chner technique, in its natural setting, is a metho d to investigate obstructions to the existence of geometric ob jects on p ositively curved compact manifolds. The technique mainly relies on the development of a suitable Laplacian identity and its analysis to explore restrictions on the existence of the ob jects under study. Following the usual Bo chner technique and the work of Eells-Sampson [7], the author develop ed a Bo chner technique for harmonic morphisms in [17 ] and presented restrictions on the existence of harmonic morphisms from p ositively curved compact Riemannian manifolds. This was further extended in [18 ] to include some non-compact domains. The conventional Bo chner technique, though very powerful, is not a handy to ol to explore restrictions on the existence of geometric ob jects on negatively curved compact domains. Until now, the investigation of general restrictions on harmonic morphisms from compact negatively curved manifolds is limited to the following cases. m n Case 1: [8] There exist no non-constant Riemannian submersions : M ! N m >n with totally geo desic bres if M has negative sectional curvature. Case 2: [2] Every harmonic morphism from the compact quotients of the hyp erb olic space 3 H to a Riemann surface is constant. Case 3: [19 ] Any non-constant submersive harmonic morphism from a compact Riemannian M n+1 n manifold M to N , such that Ric U; U 0 for U vertical, is totally geo desic. Realizing that all the maps considered ab ove are particular cases of submersive harmonic morphisms with totally geo desic bres, we take a uni ed approach and study the restrictions on the existence of harmonic morphisms with totally geo desic bres and their structure, in case these exist. In order to do so, we develop a variant of the usual Bo chner technique by develop- ing a generalized Bo chner typ e formula which leads to obtaining restrictions on the existence of harmonic morphisms, with totally geo desic, from negatively curved compact Riemannian man- ifolds. These restrictions contain the ab ove results as particular cases. A comparitive study of this generalized Bo chner typ e formula with the usual one [17 , Prop osition 2.1] provides lo cal non-existence results for harmonic morphisms, with totally geo desic bres, from non-negatively 3 curved Riemannian manifolds. As a nal consequence, we obtain a classi cation of submer- m sive harmonic morphisms, having totally geo desic bres, from op en subsets of R to complete manifolds of non-p ositive scalar curvature. A conventional remark: The sign convention adopted for the curvature is the one that coincides with the classical curvature tensor i.e. for vector elds X , Y , the curvature R of a connection r is RX; Y =r r + r r + r : X Y Y X [X;Y ] 2. Harmonic morphisms The formal theory of harmonic morphisms between Riemannian manifolds b egan with the work of Fuglede [9 ] and Ishihara [14 ]. m n De nition 2.1. A smo oth map : M ! N between Riemannian manifolds is called a harmonic morphism if, for every real-valued function f which is harmonic on an op en subset U 1 1 of N with U non-empty, f is a harmonic function on U . Harmonic morphisms are related to horizontally weakly conformal maps which can b e de- ned in the following manner. m n For a smo oth map : M ! N , let C = fx 2 M jrankd < ng be its critical set. The x p oints of the set M n C are called regular points. For each x 2 M n C , the vertical space at x is de ned by V = Kerd . The horizontal space H at x is given by the orthogonal complement x x x of V in T M . x x m n De nition 2.2. A smo oth map :M ; g ! N ; h is called horizontal ly weakly conformal if d =0 on C and the restriction of to M n C is a conformal submersion, that is, for each H M ! T N is conformal and surjective. This means that x 2 M n C , the di erential d : T x x x + there exists a function : M n C ! R such that 2 H M: hdX ;dY = g X; Y 8X; Y 2 T x + By setting =0 on C ,we can extend : M ! R to a continuous function on M such that 0 + 2 is smo oth. The extended function : M ! R is called the dilation of the map. 0 2 2 2 Let grad and grad denote the horizontal and vertical pro jections of grad . H V m n De nition 2.3. A smo oth map : M ! N is called horizontal ly homothetic if it is a horizontally conformal submersion whose dilation is constant along the horizontal curves i.e. 2 grad =0. H m n Recall that a map : M ! N is said to b e harmonic if it extremizes the asso ciated energy R 1 2 M k k d for every compact domain M . It is well-known that a map integral E = 2 is harmonic if and only if its tension eld vanishes. 4 Harmonic morphisms can b e viewed as a sub class of harmonic maps in the light of the following characterization, obtained in [9 , 14 ]. A smooth map is a harmonic morphism if and only if it is harmonic and horizontal ly weakly conformal. The following result of Baird-Eells [3 , Riemannian case] and Gudmundsson [12 , semi-Riemannian case] re ects a signi cant geometric feature of harmonic morphisms. m n Theorem 2.4. Let : M ! N bea horizontal ly conformal submersion with dilation . If 1. n =2, then is a harmonic map if and only if it has minimal bres. 2. n 3, then two of the fol lowing imply the other, a is a harmonic map b has minimal bres c is horizontal ly homothetic. The notion of horizontally conformal maps is a natural generalization of Riemannian submer- sions. The fundamental equations of Riemannian submersions were generalized for horizontally conformal submersions in [11 ]. We state those results which will b e needed in the pro of of the Bo chner formula and refer the reader to [11 ] for complete details. If T and A denote the standard fundamental tensors of a horizontally conformal submersion then the relation of the integrabilityof horizontal distribution with the tensor A is given by the following expression. 1 1 2 2.1 V [X; Y ] g X; Y grad X; Y horizontal: A Y = X V 2 2 Moreover, the mixed sectional curvatures of the domain satisfy the following relation. m n Prop osition 2.5. [11 ] Let :M ; g ! N ; h be a horizontal ly conformal submersion with total ly geodesic bres. If X , Y are horizontal vectors and U , V are vertical vectors then M 2.2 g R X; U Y; V = g r A Y; V +g A U; A V U X X Y 1 2 + g A Y; U g grad ;V X V 2 where is the dilation. For the fundamental results and prop erties of harmonic morphisms, the reader is referred to [1 , 6, 9, 20 ] and for an up dated online bibliography to [13 ]. 3. The Bochner type formula In this section we develop the generalized Bo chner typ e formula, which will be the main to ol in the next section.
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