NON-VANISHING COMPLEX VECTOR FIELDS AND THE EULER CHARACTERISTIC 9 HOWARDJACOBOWITZ 0 0 2 Abstract. Everymanifoldadmitsanowherevanishingcomplexvector field. n If, however, themanifoldiscompact andorientable andthe complexbilinear a form associated to a Riemannian metric is never zero when evaluated on the J vector field,thenthemanifoldmusthavezeroEulercharacteristic. 7 ] One of the oldest and most basic results in global differential topology relates G the topology of a manifold to the zeros of its vector fields. Let M be a compact D and orientable manifold and let χ(M) denote its Euler characteristic. Here is the h. simplest statement of this relation. t a (1) If there is a global nowhere zero vector field on M then χ(M)=0. m This of course is for a real vector field (that is, for a section M → TM). On the [ other hand, it is easy to see that any manifold admits a nowhere zero complex vector field. (A complex vector field is a section M →C⊗TM). This can be seen 1 v most simply by observing that a generic perturbation of any section, even the zero 3 section itself, must be everywhere different from zero. 9 It is natural to seek a condition on a nowhere zero complex vector field which 8 would again imply χ(M)=0. Curiously, a trivial restatement of (1) leads to such 0 a condition. Let g be any Riemannian metric on M. . 1 0 (2) Let v : M → TM be a global vector field on M. If the Riemannian metric 9 g(v,v) is never zero, then χ(M)=0. 0 : v Here is the condition for complex vector fields. i X Theorem 1. Let v : M → C⊗TM be a global vector field on M. If the bilinear r form g(v,v) is never zero, then χ(M)=0. a Here g is extended to complex vector fields by taking g(v,w) to be complex linear in each argument; for v =ξ+iη we have g(v,v)=g(ξ,ξ)−g(η,η)+2ig(ξ,η). Proof. We show that if g(v,v) 6= 0 then v can be deformed to a nowhere zero real vector field. So the Euler characteristic would be zero, according to (1). We decompose M as M =A+∪B∪A− where g(ξ,ξ) > g(η,η) on A+, the opposite inequality holds on A−, and equality holds on B. We assume for now that B is not empty. Note that ξ is nowhere zero in A+ and η is nowhere zero in A−. Further, since g(v,v) is never zero, we have 2000 Mathematics Subject Classification. Primary57R25;secondary57R20. SubmittedforpublicationJuly25,2008. 1 2 HOWARDJACOBOWITZ that g(ξ,η) is never zero on B. Thus there is an open neighborhood Ω of B on which g(ξ,η) is never zero. We may take Ω to have a smooth boundary. We have that η is never zero in A−∪Ω and ξ is never zero in A+ ∪Ω. Let Ω1 be an open set chosen so that B ⊂Ω1, Ω1 ⊂Ω and Ω1 is a neighborhood retract of Ω. The boundary of Ω has two components, one in A+ and the other in A−. (That is,the boundaryofΩ isthe unionoftwosets,neitherofwhichneedbe connected.) The same is true for the boundary of Ω1. We will work only with the components in A+. Call them Σ and Σ1. Each of these sets separates M into two components. We seek to deform v to a nowhere vanishing real vector field u. Set u = ξ on the componentofM−ΣwhichdoesnotcontainA−. ThesetsΣandΣ1 boundaregion whichretractsontoΣ1. Wewanttorotateξ toη,(orto−η)astheretractiontakes Σ to Σ1. Since g(ξ,η) 6= 0, in this region, this is easily done. Pick a point in this region. The angle θ between the vectors ξ and η satisfies one of the alternatives 0≤θ ≤π/2 or π/2<θ ≤π and whichever alternative is satisfied at that point is also satisfied at all points in the region. Thus as we retract Σ to Σ1, we may rotate ξ to η, or, respectively to −η. Finally,defineu=η,respectivelyu=−η,onthe componentofM−Σ1 which contains A−. If B is empty, the proof is even easier. Now either g(ξ,ξ) > g(η,η) everywhere and so ξ is a nowhere zero real vector field or the opposite inequality holds and η is a nowhere zero real vector field. (cid:3) Remark. Wehaveprovedthetheorembyreducingto(1). Thislatterresultgoes back to H. Hopf; an influential modern proof was given by Atiyah [1]. Atiyah’s proof makes use of the Clifford algebra structure on the bundle of exterior forms. OurTheoremcanbeproveddirectly,withoutreducingto(1),byfollowingAtiyah’s proof using the corresponding complex Clifford algebra. There is a strongerversionof(1) whichexpressesthe Euler characteristicas the algebraicsumoftheindicesofthezerosofthevectorfield. (Indeedthisistheresult of Hopf.) It would be interesting to generalize this to complex vector fields. References [1] Atiyah, M., Vector fields on manifolds. Arbeitsgemeinschaft fu¨r Forschung des Landes Nordrhein-Westfalen,Heft200Westdeutscher Verlag,Cologne197026pp. Rutgers University,Camden,New Jersey 08012 E-mail address: [email protected]