* On the structure vector eld of a real hypersurface in complex quadric

The complex quadric Qm = SOm+2/SOmSO2 is a compact Hermitian symmetric space of rank 2. It is also a complex hypersurface in complex projective spaceCPm+1, [1]. Qm is equippedwith two geometric structures: a complex conjugation A and a Kähler structure J. Real hypersurfaces M in Qm are immersed submanifolds of real codimension 1. The Kähler structure J of Qm induces on M an almost contact metric structure (φ, ξ, η, g), where φ is the structure tensor eld, ξ is the structure (or Reeb) vector eld, η is a 1-form and g is the the induced Riemannian metric on M. Real hypersurfacesM in Qm whose Reeb ow is isometric are classi ed in [2]. They obtain tubes around the totally geodesic CPk in Qm when m = 2k. The condition of isometric Reeb ow is equivalent to the commuting condition of the shape operator S with the structure tensor eld φ of M. It is known that a Killing vector eld X on a Riemannian manifold (M̄, ḡ) satis es LX ḡ = 0, where L denotes the Lie derivative. Killing vector elds are a powerful tool in studying the geometry of a Riemannian manifold. A Killing vector eld is a Jacobi vector eld along any geodesic. However the converse is not true: the position vector on the euclidean space Rn is a Jacobi eld along any geodesic of Rn but it is not Killing. Studying when the structure vector eld of a complex projective space is Killing, Deshmukh, [3], introduced the notion of Jacobi type vector elds on a Riemannian manifold. A vector eld Y on M̄ is of Jacobi type if it satis es ∇̄X∇̄XY + R̄(Y , X)X = 0 (1)


Introduction
The complex quadric Q m = SO m+ SO m SO is a compact Hermitian symmetric space of rank 2. It is also a complex hypersurface in complex projective space CP m+ , [1].Q m is equipped with two geometric structures: a complex conjugation A and a K ähler structure J.
Real hypersurfaces M in Q m are immersed submanifolds of real codimension 1.The K ähler structure J of Q m induces on M an almost contact metric structure (φ, ξ, η, g), where φ is the structure tensor eld, ξ is the structure (or Reeb) vector eld, η is a 1-form and g is the the induced Riemannian metric on M.
Real hypersurfaces M in Q m whose Reeb ow is isometric are classi ed in [2].They obtain tubes around the totally geodesic CP k in Q m when m = k.The condition of isometric Reeb ow is equivalent to the commuting condition of the shape operator S with the structure tensor eld φ of M.
It is known that a Killing vector eld X on a Riemannian manifold ( M, ḡ) satis es L X ḡ = , where L denotes the Lie derivative.Killing vector elds are a powerful tool in studying the geometry of a Riemannian manifold.A Killing vector eld is a Jacobi vector eld along any geodesic.However the converse is not true: the position vector on the euclidean space R n is a Jacobi eld along any geodesic of R n but it is not Killing.Studying when the structure vector eld of a complex projective space is Killing, Deshmukh, [3], introduced the notion of Jacobi type vector elds on a Riemannian manifold.A vector eld for any vector eld X tangent to M, where ∇ denotes the Levi-Civita connection on M and R its Riemannian curvature tensor.Naturally any Jacobi type vector eld on M is a Jacobi vector eld along any geodesic of M. As on a real hypersurface M in Q m we have a special vector eld, the structure one ξ, it is interesting to see if it is Killing when it is of Jacobi type.In this sense we will prove the following Theorem 1.1.Let M be a real hypersurface in Q m , m ≥ .If M is either compact or Hopf and the structure vector eld is of Jacobi type, it is a Killing vector eld.
By this Theorem and the classi cation of real hypersurfaces with geodesic Reeb ow we obtain Corollary 1.2.Let M be a compact or Hopf real hypersurface in Q m , m ≥ .Then the structure vector eld is of Jacobi type if and only if m is even, say m = k, and M is locally congruent to a tube around a totally geodesic Similar results for real hypersurfaces of complex two-plane Grassmannians were obtained in [4].

The space Q m
For the study of Riemannian geometry of Q m see [1].All the notations we will use since now are the ones in [2].
The complex projective space CP m+ is considered as the Hermitian symmetric space of the special unitary group SU m+ , namely CP m+ = SU m+ S(U m+ U ).The symbol o=[0,...,0,1] in CP m+ is the xed point of the action of the stabilizer S(U m+ U ).The action of the special orthogonal group SO m+ ⊂ SU m+ on CP m+ is of cohomogeneity one.A totally geodesic projective space RP m+ ⊂ CP m+ is an orbit containing o.The second singular orbit of this action is the complex quadric Q m = SO m+ SO m SO .It is a homogeneous model wich interprets geometrically the complex quadric Q m as the Grassmann manifold G + (R m+ ) of oriented 2-planes in R m+ .For m = the complex quadric is isometric to a sphere S of constant curvature.For m = the complex quadric Q is isometric to the Riemannian product of two 2-spheres with constant curvature.Therefore we assume the dimension of the complex quadric Q m to be greater than or equal to 3.
Moreover, the complex quadric Q m is the complex hypersurface in CP m+ de ned by the equation z + ... + z m+ = , where z i , i = , ..., m + , are homogeneous coordinates on CP m+ .The K ähler structure of complex projective space induces canonically a K ähler structure (J, g) on Q m , where g is the Riemannian metric induced by the Fubini-Study metric of CP m+ .
A point [z] in CP m+ is the complex span of z, that is [z] = {λz λ ∈ C}, where z is a nonzero vector of The shape operator A z of Q m with respect to the unit normal vector z is given by where V(A z ) is the (+1)-eigenspace of A z and JV(A z ) is the (-1)-eigenspace of A z .Geometrically, it means that A z de nes a real structure on the complex vector space T [z] Q m .The set of all shape operators A λz de nes a parallel rank 2 subbundle A of the endomorphism bundle End(T Q m ) which consists of all the real structures of the tangent space of Q m .For any A ∈ A, A = I and AJ = −JA.
The Gauss equation of Q m in CP m+ yields that the Riemannian curvature tensor R of Q m is given by where J is the complex structure and A is a real structure in A.
For every vector eld W tangent to Q m there is a complex conjugation A ∈ A and orthonormal vectors

Real hypersurfaces in Q m
Let M be a real hypersurface in Q m , that is, a submanifold of Q m with real codimension one.The induced Riemannian metric on M will also be denoted by g, and ∇ denotes the Riemannian connection of (M, g).Let N be a unit normal vector eld of M and S the shape operator of M with respect to N. For any X tangent to M we write JX = φX + η(X)N where φX denotes the tangential component of JX and η(X)N its normal component.The structure vector eld (or Reeb vector eld) ξ is de ned by ξ = −JN.The 1-form η is given by η(X) = g(X, ξ) for any vector eld X tangent to M. Therefore, on M we have an almost contact metric structure (φ, ξ, η, g).Thus, for all tangent vector elds X, Y on M.Moreover, the parallelism of J yields and for any X, Y tangent to M.
Let X ∈ T [z] M. Then AX is decomposed into where BX is the tangential component of AX and ρ(X)N is its normal component, with ρ(X) = g(AX, N).As seen above ρ(ξ) = .
From (2) the curvature tensor R of M is given by for any X, Y , Z tangent to M, where (.) T denotes the tangential component of the correspondent vector eld.From (8) the Ricci tensor of M is given (see [5]) by for any X tangent to M.Moreover, the Codazzi equation is given by for any X, Y , Z tangent to M.
The real hypersurface M is called Hopf if the Reeb vector eld is an eigenvector of the shape operator S, that is Sξ = αξ where α = g(Sξ, ξ) is the Reeb function.

Proof of Theorem 1.1
Let us suppose that ξ is of Jacobi type.Then ∇ X ∇ X ξ + R(ξ, X)X = for any X tangent to M. Take an orthonormal basis {e , ..., e m− } of vector elds tangent to M. As ξ is of Jacobi type, From ( 9) From ( 11) and ( 12) we get ∑ m− i= Taking its scalar product with ξ and bearing in mind that g(Bξ, ξ) = g(Aξ, ξ) we obtain From (13) and Lemma 4.2 we obtain Bearing in mind (17) Codazzi equation yields In both cases as (L ξ g)(X, Y) = g((φS−Sφ)X, Y), for any X, Y ∈ TM, we conclude L ξ g = and ξ is Killing, obtaining our Theorem.
As φS = Sφ we have, [2], that m = k and M must be locally congruent to a tube around a totally geodesic Bearing in mind the expression of the shape operator S of such a real hypersurface, [2], it is immediate to see that its structure vector eld is of Jacobi type and we conclude the proof of our Corollary.