New aspects on CR-structures of codimension 2 on

New aspects on CR-structures of codimension 2 on hypersurfaces of Sasakian manifolds ∗ by Marian-Ioan Munteanu Abstract. We introduce a torsion free linear connection on a hypersurface in a Sasakian manifold on which we have defined in natural way a CR-structure of CR-codimension 2. We study the curvature properties of this connection and we give some interesting examples. Mathematics Subject Classification (2000): 53C25, 53C15, 53B15. Keywords and Phrases: CR−structures, almost contact structures, f -structure with complemented frames. 1. Introduction In 1995, P.Matzeu & V.Oproiu have introduced a torsion free linear connection adapted to an almost contact structure associated with a given pseudoconvex CRmanifold (of hypersurface type) (see [5]). The fundamental tensor field and the 1form of the associated almost contact structure are no longer parallel with respect to this connection. Yet, this connection yields to the same Bochner type curvature tensor field for the CR-manifold as it was obtained by using the Tanaka Webster connection. In this paper we consider CR-structures of codimension 2 on hypersurfaces in Sasakian manifolds. We use a natural f -structure with complemented frames in order to obtain a torsion free linear connection. This is a generalization of the Matzeu Oproiu connection for the CR-codimension 2. Then, we give a relation between the adapted connection and the Levi Civita connection on the hypersurface. Finally, we examine some symmetry properties of the curvature tensor field of this connection. In the end of the paper we present some examples in the case when the ambient is R5 and S 5 endowed with the canonical Sasakian structures. This work was partially supported by Grant 28/100/2003 MEC. ∗ A part of this paper was presented at the 9-th International Conference on Differential Geometry and Its Applications, Prague, 2004. 1 ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 2 2. The adapted torsion free canonical connection f be a smooth Sasakian manifold of dimension 2n + 3 with the contact metric Let M e ξ, e ηe, ge) and let ∇ e be its Levi-Civita connection. The following relation structure (φ, e = −e e X φ)Y f), (∇ g (X, Y )ξ + ηe(Y )X, for X, Y ∈ χ(M (2.1) f and actually characterizes Sasakian manifolds among almost contact holds on M f tangent 1 to Riemannian manifolds. Let M be an oriented C ∞ hypersurface in M f the immersion of M in M f. On M f the structure vector field ξe and let ι : M ֒→ M we have e ) + ηe(X)e f) ge(X, Y ) = de η (X, φY η (Y ) , ∀ X, Y ∈ χ(M (2.2) (if ω is a 1-form, recall the formula: 2dω(X, Y ) = Xω(Y ) − Y ω(X) − ω([X, Y ])). On M we set the 1-form η = ι∗ ηe. (2.3) Let ξ be the restriction of ξe to M (since the last one is tangent to the hypersurface ξ is tangent to M ) and consequently we have ι∗ ξ = ξe . (2.4) f) the unit vector field normal to M we can define U ∈ If we denote by N ∈ χ(M χ(M ) such that e ι∗ U = φN (2.5) e is a vector field tangent to M ). Let g be the induced metric (since φN and define the 1-form u on M by g = ι∗ ge . u(X) = g(U, X) , ∀ X ∈ χ(M ) . (2.6) (2.7) Consider the distribution H(M ) = {X ∈ χ(M ) : η(X) = 0 , u(X) = 0} and the endomorphism J : H(M ) −→ H(M ) given by the restriction of φe to H(M ), i.e. ι∗ JX = φe ι∗ X . (2.8) 1 Many geometers use to consider ξ tangent to the submanifold because in the the theory of CR submanifolds the condition M normal to ξ leads to M anti-invariant submanifold (see [7], Proposition 1.1, p. 43) and the condition ξ oblique gives very complicated embedding equations. ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 3 It can be proved that the definition is good (in the following sense: if X ∈ H(M ) then JX ∈ H(M )). Moreover, J has the property J 2 = −idH(M ) . (2.9) The tangent space of M can be decomposed in the following direct sum T (M ) = H(M ) ⊕ span [U ] ⊕ span [ξ] . (2.10) Let us remark that dη is non degenerate on H(M ). Lemma 2.1 We have ∞ [ξ, Γ(H(M ))] ⊂ Γ(H(M )) (2.11) [U, Γ(H(M ))] ∈ ker η, (2.12) where Γ(H(M )) is the C (M )-module of smooth sections in H(M ). Proof. It is obvious that [ξ, Γ(H(M ))] ⊂ ker η. Let’s compute u([ξ, X]) where X ∈ H(M ). We have e . e eX − ∇ e X ξ) u([ξ, X]) = g(U, [ξ, X]) = ge(U, ∇ ξ (Sometimes we will give up the notation ι∗ for vector fields tangent to M even that e If f.) We know that ∇ e X ξe = φX. these will be thought as vector fields tangent to M e belongs to H(M ) too, thus it is orthogonal to U . We X belongs to H(M ) then φX e obtain, since ∇ξe U = −N that e eX) = −e e e U, X) = 0 u([ξ, X]) = ge(U, ∇ g (∇ ξ ξ which means that [ξ, X] ∈ ker u. Thus [ξ, X] ∈ H(M ) for all X ∈ H(M ). In order to prove (2.12) we will show that dη(U, X) = 0 for all X ∈ Γ(H(M )). We have dη(U, X) = (ι∗ de η )(U, X) = de η (ι∗ U, ι∗ X) = ge(φ̃ 2 N, ι∗ X) = −e g (N, ι∗ X) = 0 (since ι∗ X is tangent to M while N is normal). Proposition 2.2 The vector fields ξ and U are orthogonal and of norm 1. Moreover [U, ξ] = 0 . (2.13) ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 4 Proof. We will prove only the second part of this Proposition. Since the inner product ξydη is zero one gets [U, ξ] ∈ ker η. Then we use the same technique as above: e ) − ge(U, ∇ eUξ − ∇ e ξ U ) = ge(U, φU e ξ U ). u([U, ξ]) = g(U, [U, ξ]) = ge(U, ∇ e are orthogonal (with respect to ge) and since ||U || = 1, by derivation But U and φU e ξ U ) = 0. It follows that [U, ξ] ∈ ker u. with respect to ξ one gets ge(U, ∇ Consequently [U, ξ] ∈ H(M ). We have to compute now the H(M )-component. To do this we will use the non degeneracy of dη on H(M ). For an arbitrary X ∈ H(M ) we have 2dη([U, ξ], X) = −η([[U, ξ], X]). In the Jacobi identity [[U, ξ], X] + [[ξ, X], U ] + [[X, U ], ξ] = 0 applying η and by taking into account the Lemma 2.1 one obtains η([[U, ξ], X]) = 0. We get the conclusion. Proposition 2.3 The distribution H(M ) defines a CR-structure on M of CRcodimension 2. Proof. Let X, Y ∈ H(M ). We have to verify the two integrability conditions in order to obtain a CR-structure, namely a) [JX, JY ] − [X, Y ] ∈ H(M ), b) NJ (X, Y ) = 0, where NJ is the Nijenhuis tensor of J. An easy computation yields to a). Then, from the normality condition of the e ξ, e ηe, ge) on M f, i.e. N e + 2de η ⊗ ξe = 0, one gets ι∗ NJ (X, Y ) = 0 Sasakian structure (φ, φ (for X, Y ∈ Γ(H(M ))). In the next we define on M a tensor field f of type (1, 1) as follows f : χ(M ) −→ χ(M ) f X = JX for X ∈ H(M ) , (2.14) f U = 0 , f ξ = 0. With respect to the decomposition (2.10), if X is an arbitrary vector field on M then it can be written in the form X = XH(M ) + u(X) U + η(X)ξ where XH(M ) is the H(M ) component of X. Thus f verifies f3 + f = 0 . (2.15) ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 5 Proposition 2.4 The structure (φ, ξ, U, η, u) defined on M is an f structure with complemented frame (see S.I.Goldberg & K.Yano, [3]) or, in other terminology, an f structure with parallelizable kernel (f -pk structure), i.e. η(ξ) = 1 , η(U ) = 0 , u(ξ) = 0 , u(U ) = 1 fξ = 0 , fU = 0 , η ◦ f = 0 , u ◦ f = 0 f 2 = −I + η ⊗ ξ + u ⊗ U . If X and Y are vector fields on M then we have g(X, Y ) = dη(X, f Y ) + u(X)u(Y ) + η(X)η(Y ) . (2.16) In [4] it is defined a linear connection on an almost S-manifold which generalizes the Tanaka-Webster connection for strictly pseudoconvex CR-manifolds of hypersurface type. In the following we will construct a torsion free connection on M as being the analogue of Matzeu-Oproiu connection. From now on we will suppose that the inner product U y du = 0 (2.17) holds on M ; this condition yields to [U, Γ(H(M ))] ⊂ ker u and consequently, by virtue of (2.12) we obtain [U, Γ(H(M ))] ⊂ Γ(H(M )). The condition above is a weaker condition than the ”S” condition du = Φ (here ˙ U U vanishes, Φ(X, Y ) = g(f X, Y )) and it is equivalent to: H(M ) component of ∇ ˙ where ∇ is the Levi Civita connection of g. Moreover, this means that H(M ) component of AU vanishes too, where A is the Weingarten operator. There are some important cases in which this happens, namely f, i.e. • (TCG) M is a totally contact geodesic submanifold in M B(X, Y ) = ηe(X)B(Y, ξ) + ηe(Y )B(X, ξ) f, i.e. • (TCU) M is a totally contact umbilical submanifold in M B(X, Y ) = [e g (X, Y ) − ηe(X)e η (Y )] α + ηe(X)B(Y, ξ) + ηe(Y )B(X, ξ) where α is a vector field normal to M ; it follows that α = λN with λ non-vanishing f) smooth function on M (B is the second fundamental form of M in M f, i.e. • (PUH) M is a pseudo umbilical hypersurface in M AX = λ (X − η(X)ξ) + µ u(X) U − η(X)U − u(X) ξ with λ, µ ∈ C ∞ (M ). ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 6 Let us remark that in the case (TCG) the 1-form u is closed, while in cases (TCU) and (PUH) we have du(X, Y ) = λg(X, f Y ) which, in general, is different from 0. Recall two formulas (Gauss and Weingarten): e XY = ∇ ˙ X Y + b(X, Y )N , X, Y ∈ χ(M ) (G) ∇ e X N = −AX, X ∈ χ(M ) (W) ∇ where b(X, Y ) is the scalar second fundamental form. Since the ambient is Sasakian one gets: ˙ X ξ = f X , b(X, ξ) = −u(X) (1) ∇ ˙ X U = −f AX , b(X, U ) = u(AX) (2) ∇ ˙ (3) ∇U U = 0 (4) η(AX) = −u(X) , u(AX) = −η(X) + b u(X), where b = b(U, U ) (5) Aξ = −U , AU = −ξ + b U , AX ∈ H(M ) if X ∈ H(M ) (6) AX = −η(X)U − u(X)ξ + b u(X)U + H(M ), with X ∈ χ(M ). Denote by ψ = 12 (Lξ f ) and p = 21 (LU f ) where L is the Lie derivative. If the f is only a contact manifold (not necessarily Sasakian) we give ambient manifold M Proposition 2.5 The following statements hold: 1) ψξ = 0 , ψU = 0 , pξ = 0 , pU = 0 2) f ψ + ψf = 0 , f p + pf = 0 3) η ◦ ψ = 0 , η ◦ p = 0 , u ◦ ψ = 0 , u ◦ p = 0 4) dη(ψX, Y ) = −dη(X, ψY ) , dη(pX, Y ) = −dη(X, pY ) 5) the operators ψ and p are self-adjoint with respect to the metric g, i.e. g(ψX, Y ) = g(X, ψY ) , g(pX, Y ) = g(X, pY ). Proof. The relations 1) - 3) are immediately. Let’s prove the first relation in 4) (the second can be proved in the same way). Let X, Y ∈ H(M ). Then 2dη(ψX, Y ) = ξ(dη(f X, Y )) − dη([X, ξ], f Y ) − dη([Y, ξ], f X) . Interchanging X and Y we get dη(ψY, X) = dη(ψX, Y ) and hence the conclusion. The last statement can be obtained easily from 1) - 4). f is Sasakian. In our case M Proposition 2.6 We have 1) ψ ≡ 0 2) 2pX = (A + f Af )X + u(X) ξ + η(X) U − b u(X) U ; p vanishes in cases (TCG), (TCU) and (PUH). ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 7 f is Sasakian). Proof. 1) is a direct consequence of Lξe φe = 0 (since M For 2) the following relations hold: ³ ´ e + u(X) N ) + f Af X = ˙ U (f X) − ∇ ˙ f X U = tan ∇ e U (φX [U, f X] = ∇ e U X) = η(X) U + f Af X + tan(φe∇ ˙ U X − AX + η(AX) ξ + u(AX) U . f [U, X] = f ∇ Now, by using (G) and (W) we get the statement. f then b = 0; if M is TCU then b = λ and if M is PUH then If M is TCG in M b = λ + µ. One gets p = 0. In the following we are looking for a torsion free connection on M related with the structures defined so far. We can state the following theorem. Theorem 2.7 There exists one and only one torsion free connection on M such that (∇X η)(Y ) = dη(X, Y ) , ∇X dη = 0 , ∇X ξ = 0 , (∇X u)(Y ) = du(X, Y ) ∇X U = 0 (2.18) (∇X f )Y = u(X) {(A + f Af )Y + u(Y )ξ + η(Y )U − b u(Y )U } − −dη(X, f Y )ξ + dη(AX, f Y )U. f is only a contact manifold the last condition in (2.18) must be Remark 2.8 If M replaced by (∇X f )Y = 2η(X)ψY + 2u(X)pY − dη(X, f Y )ξ − du(X, f Y )U (2.19) which generalizes the fourth condition in (3.1) from [5] pg. 5. Remark 2.9 If M is TCG, TCU or PUH then du = −λ dη (λ can vanish) and the last condition in (2.18) becomes (∇X f )Y = dη(f X, Y ) ζ where ζ = ξ − λ U . Proof. (of Theorem 2.7) Before starting with the proof of the theorem let us remark that the conditions in (2.18) have been suggested by (3.1) from [5]. We also assumed the compatibility conditions with the structure defined so far. To get the connection ∇ (in terms of the f -pk structure) we shall compute η(∇X Y ), u(∇X Y ) and dη(∇X Y, Z) for X, Y, Z ∈ χ(M ). ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 8 We obtain immediately η(∇X Y ) = X(η(Y )) − dη(X, Y ) (2.20) u(∇X Y ) = X(u(Y )) − du(X, Y ) . Let’s compute now ∇g. First we have (∇X g)(Y, Z) = (∇X dη)(Y, f Z) + dη(Y, (∇X f )Z)+ +(∇X η)(Y )η(Z) + η(Y )(∇X η)(Z)+ (2.21) +(∇X u)(Y )u(Z) + u(Y )(∇X u)(Z) . Taking into account (2.18) we get (∇X g)(Y, Z) = u(X) dη(Y, (A + f Af )Z)+ (2.22) +dη(X, Y )η(Z) + dη(X, Z)η(Y ) + du(X, Y )u(Z) + du(X, Z)u(Y ) . Let us remark that if we restrict to the distribution H(M ) we have (∇g)|H(M ) = 0 . (2.23) In the following we will do similar computations as in the Levi Civita theorem (since the connection is torsion free). We can write 2g(∇X Y, Z) = Xg(Y, Z) + Y g(Z, X) − Zg(X, Y )+ +g([X, Y ], Z) − g([X, Z], Y ) − g([Y, Z], X)− (2.24) −(∇X g)(Y, Z) − (∇Y g)(X, Z) + (∇Z g)(X, Y ) . Since dη(∇X Y, Z) = −g(∇X Y, f Z) for all X, Y, Z ∈ χ(M ) we obtain 2dη(∇X Y, Z) = Xdη(Y, Z) + Y dη(X, Z) + (f Z)dη(X, f Y )+ +dη([X, Y ], Z) + dη([X, f Z], f Y ) + dη([Y, f Z], f X)+ +u(X)dη(Z, (Af − f A)Y ) + u(Y )dη(Z, (Af − f A)X). (2.25) ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 9 f is only a contact manifold the relation before becomes If M 2dη(∇X Y, Z) = Xdη(Y, Z) + Y dη(X, Z) + (f Z)dη(X, f Y )+ +dη([X, Y ], Z) + dη([X, f Z], f Y ) + dη([Y, f Z], f X)+ (2.26) +2η(X)dη(f Z, ψY ) + 2η(Y )dη(f Z, ψX)+ +2u(X)dη(f Z, pY ) + 2u(Y )dη(f Z, pX) . The relations (2.20) and (2.25) completely define the connection ∇ (since dη is non degenerate on H(M )). Furthermore, if ∇′ is another connection satisfying the hypotheses of the theorem, we have η(∇X Y ) = η(∇′X Y ), u(∇X Y ) = u(∇′X Y ) and dη(∇X Y, Z) = dη(∇′X Y, Z) which imply ∇ = ∇′ . We are interested now to find a relation between the adapted connection and the ˙ Define and endomorphism S on H(M ) by Levi Civita connection ∇. dη(SX, Y ) = −du(X, f Y ) , X, Y ∈ H(M ) (2.27) (due the non degeneracy of dη the endomorphism S is well defined). An easy computation yields to du(X, Y ) = 12 (g(AX, f Y ) − g(AY, f X)), for all X, Y ∈ χ(M ). Consequently we obtain S = − 12 (f A + Af ). We can extend S, if necessary, to span[ξ] ⊕ span[U ] by putting Sξ = 0 and SU = 0 and the previous formula remains true. Taking into account that the difference between two torsion free connections is a symmetric (1, 2) tensor field, after a straightforward computations one gets ˙ = α ⊗ U + 2 (u ⊙ f A − η ⊙ f ) ∇−∇ (2.28) where with α(X, Y ) = 1 2 (α ⊗ U ) (X, Y ) = α(X, Y )U (g(AX, f Y ) + g(AY, f X)) and ⊙ is the symmetric product, i.e. (η ⊙ f ) (X, Y ) = and (η ⊙ f A) (X, Y ) = for all X, Y ∈ χ(M ). 1 (η(X)f Y + η(Y )f X) 2 1 (η(X)f AY + η(Y )f AX) 2 ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 10 f is only a contact manifold the relation between ∇ and ∇ ˙ is Remark 2.10 If M given by ˙ = α ⊗ U + 2 (η ⊙ h + u ⊙ k − u ⊙ S − η ⊙ f ) ∇−∇ (2.29) where h = f ψ and k = f p. (The equivalence between (2.28) and (2.29) holds due the fact that on H(M ) we have LU f = A + f Af if the ambient manifold is Sasakian.) Remark 2.11 If M is TCG, TCU or PUH then α = 0 and ˙ = −2 θ ⊙ f ∇−∇ where θ = η − λ u. 3. Curvature of the torsion free adapted connection Consider the curvature tensor field R of ∇ defined by RXY Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ] Z , X, Y, Z ∈ χ(M ). (3.1) We will find some general relations and properties of R and especially for the restriction of R on H(M ). Equation (2.18) imply: RXY ξ = 0 , RXY U = 0 , ∀ X, Y ∈ χ(M ). (3.2) Moreover, RXY Z belongs to ker η but it is not necessarily a section in H(M ) for all X, Y, Z ∈ χ(M ). Yet, RU X Y ∈ Γ(H(M )). We also have RXY f Z = f RXY Z + 4du(X, Y )pZ− −(∇X du)(Y, f Z) U + (∇Y du)(X, f Z) U , X, Y, Z ∈ Γ(H(M )). (3.3) Define now a 4 covariant tensor field R by R(W, Z, X, Y ) = g(W, RXY Z) , X, Y, Z ∈ χ(M ) (3.4) (R is a kind of Riemann Christoffel tensor). We are interested now to find some symmetry properties for the tensor field R similar those for the usual Riemann Christoffel tensor (in Riemannian geometry). Obviously R(W, Z, X, Y ) = −R(W, Z, Y, X) and S R(W, Z, X, Y ) = 0 (due (X,Y,Z) to the first Bianchi identity fulfilled by R). Using (3.3) we get R(Z, Z, X, Y ) = dη(Z, AZ) (dη(X, AY ) − dη(Y, AX)) (3.5) ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 11 (with X, Y, Z ∈ Γ(H(M ))) which implies R(Z, W, X, Y ) + R(W, Z, X, Y ) = = (dη(Z, AW ) + dη(W, AZ))(dη(X, AY ) − dη(Y, AX)). (3.6) As consequence we have Proposition 3.1 The Riemann Christoffel tensor field R of the linear connection ∇ satisfies the following equation R(W, Z, X, Y ) − R(X, Y, W, Z) = = dη(X, AY )dη(Z, AW ) − dη(Y, AX)dη(W, AZ) + dη(W, AX)dη(Z, AY ) −dη(X, AW )dη(Y, AZ) + dη(X, AZ)dη(Y, AW ) − dη(Z, AX)dη(W, AY ). (3.7) From the relation above we easily obtain R(E, Z, E, Y ) − R(E, Y, E, Z) = dη(E, AE) (dη(Z, AY ) − dη(Y, AZ)) (3.8) where E, Y, Z are sections in H(M ). Now, replacing E by f E and taking into account that dη(f E, Af E) = −dη(E, AE) we get R(E, Y, E, Z) + R(f E, Y, f E, Z) = R(E, Z, E, Y ) + R(f E, Z, f E, Y ). (3.9) f then R is skew-symmetric in first Remark 3.2 If M is TCG, TCU or PUH in M two arguments and pairs symmetric (i.e. R(W, Z, X, Y ) = R(X, Y, W, Z)). Consider the two times covariant tensor ρ(R)(Y, Z) = trace (X 7−→ RXY Z) , X, Y, Z ∈ χ(M ) (3.10) (where the trace is made by using the metric g) – the tensor defined above is a kind of Ricci tensor. If we take an orthonormal basis of the form {Ei , f Ei , ξ, U }i=1,n on M , the Ricci tensor can be written as ρ(R)(Y, Z) = n X i=1 {R(Ei , Z, Ei , Y ) + R(f Ei , Z, f Ei , Y )} . As consequence of the relation (3.9) we have the symmetry of the Ricci tensor, namely ρ(R)(Y, Z) = ρ(R)(Z, Y ) , Y, Z ∈ Γ(H(M )). (3.11) Moreover, we have ρ(R)(f Y, f Z) − ρ(R)(Y, Z) = 4(du(pY, f Z) + du(pZ, f Y )). (3.12) ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 12 We will say that the CR-manifold is CR-Einstein if ρ(R)(X, Y ) = λ g(X, Y ) for all X, Y ∈ H(M ), where λ ∈ C ∞ (M ). We will end this section by studying the situation AX = λ(X − η(X)ξ) + µu(X) U − η(X) U − u(X)ξ with λ and µ not necessarily non vanishing smooth functions on M . This case, let’s call it (λ − µ) contains the three cases TCG, TCU and PUH. ˙ X f )Y = −g(X, Y ) ζ for X, Y ∈ H(M ). We have dη(X, AY ) = λdη(X, Y ) and (∇ Consequently, the most of the relations involving the curvature tensor fields simplifies. For example we have for X, Y, Z, W ∈ H(M ) RXY (f Z) = f RX,Y Z − {X(λ)dη(Y, f Z) − Y (λ)dη(X, f Z)} U R(Z, W, X, Y ) = −R(W, Z, X, Y ) , R(W, Z, X, Y ) = R(X, Y, W, Z). These yield to R(f W, f Z, X, Y ) = R(W, Z, X, Y ) , R(f W, f Z, f X, f Y ) = R(W, Z, X, Y ) and hence ρ(R)(f Y, f Z) = ρ(R)(Y, Z). Let write now the relation between the curvature tensor R (of the adapted connection) and Ṙ (the curvature tensor of the Levi Civita connection) - in general this relation is very complicated: ṘXY Z = RXY Z + 2(1 + λ2 )dη(X, Y )f Z + (u(X)Y (λ) − u(Y )X(λ))f Z− −(θ(X)dη(f Y, Z) − θ(Y )dη(f X, Z))ζ + θ(Z)(θ(X)f 2 Y − θ(Y )f 2 X)− (3.13) −((1 + λ2 )dη(Y, Z) − Y (λ)u(Z))f X + ((1 + λ2 )dη(X, Z) − X(λ)u(Z))f Y. If we consider W = X = E, with E ∈ H(M ) and of norm 1, one gets Ṙ(E, Z, E, Y ) = R(E, Z, E, Y ) + θ(Y )θ(Z) − 3(1 + λ2 )g(E, f Y )g(E, f Z)− −(u(Z)g(E, f Y ) + u(Y )g(E, f Z)) dλ(E). It follows that ρ(Ṙ)(Z, Y ) = ρ(R)(Z, Y ) − 2(1 + λ2 )g(f Y, f Z)+ +2n θ(Y )θ(Z) − (u(Y )dλ(f Z) + u(Z)dλ(f Y )). ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 13 4. Examples In this section we will give some examples. Let consider as ambient manifolds R5 with (global) coordinates x, y, v, w, z. On R5 we have the usual Sasakian structure as follows:  ∂ ηe = 12 (dz − ydx − wdv) , ξe = 2 ∂z      ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ (4.1) = ∂y , φe ∂y = − ∂x − y ∂z , φe ∂v = ∂w , φe ∂w = − ∂v − w ∂z φe ∂x      ge = ηe ⊗ ηe + 41 (dx2 + dy 2 + dv 2 + dw2 ) If the hypersurface M is given by f (x, y, v, w, z) = 0 (where f is a smooth function on R5 ) the tangency condition of the structure vector field ξe to M yields to the fact that f does not depend on z. After the computations we obtain the expression of ∂ the gradient of f . We will denote ∂x by ∂x and similarly for the other coordinates. With these notations,³ the unitary vector field N normal³ to the hypersurface M ´ ´ ∂f ∂f ∂f ∂f ∂f ∂f 2 is given by N = µ ∂x ∂x + ∂y ∂y + ∂v ∂v + ∂w ∂w + y ∂x + w ∂v ∂z where 1 µ = (fx2 + fy2 + fv2 + fw2 ) 2 . Hence we obtain the expression of U : U= 2 (−fy ∂x + fx ∂y − fw ∂v + fv ∂w − (yfy + wfw ) ∂z ) . µ (4.2) Remark that [U, ξ] = 0. A vector field X ∈ χ(R5 ) is tangent to M and belongs to ker η if it is of the following form X = A(∂x + y∂z ) + B∂y + C(∂v + w∂z ) + D∂w (4.3) where A, B, C, D are smooth functions on M satisfying Afx + Bfy + Cfv + Dfw = 0 . (4.4) Then, if we ask X ∈ ker u we get another condition, namely −Afy + Bfx − Cfw + Dfv = 0 . (4.5) Since µ 6= 0 we may suppose that fv2 + fw2 6= 0, otherwise we have fx2 + fy2 6= 0 and the computations are similar. Let’s make some notations a = fx fw − fy fv , b = fx fv + fy fw , α = fv2 + fw2 . By using the relations (4.4) and (4.5) we obtain C= 1 α (aB − bA) , D = − α1 (aA + bB) . (4.6) ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 14 Consequently, we obtain a basis in H = ker η ∩ ker u   X1 = α∂x − b∂v − a∂w + (yα − wb)∂z  (4.7) X2 = α∂y + a∂v − b∂w + wa∂z = JX1 where J is the restriction of φe to H. Let us remark that |X1 | = |X2 | = µ 2 √ α. Example 1. (TCG) Consider M the hyperplane (passing by the origin and being parallel with z axis) defined by f (x, y, z, v, w) = ax + by + cv + dw ≡ 0 where a, b, c, d ∈ R with a2 + b2 + c2 + d2 = µ2 6= 0. We have U= 2 µ {−b∂x + a∂y − d∂v + c∂w − (by + dw) ∂z } and X1 = (c2 + d2 ) ∂x − (ac + bd) ∂v − (ad − bc) ∂w + [(c2 + d2 )y − (ac + bd)w] ∂z . The 2-form du vanishes identically and we have [U, X1 ] = 0, [U, JX1 ] = 0. Com2 2 2 puting the connection ∇ we obtain ∇X1 X2 = − µ (c 4+d ) ξ . As consequence, the connection is flat. Example 2. (TCU) Let M be defined by f (x, y, v, w, z) = x2 +y 2 +v 2 +w2 −1 ≡ 0 (a hyper cylinder S 3 × R in R5 ). The vector field U is given by U = 2{x∂y − y(∂x + y∂z ) + v∂w − w(∂v + w∂z )} . Moreover, we have µ = 2 , a = 4(xw − yv) , b = 4(xv + yw) , α = 4(v 2 + w2 ) and consequently X1 is determined by X1 = 4{(v 2 + w2 )∂x − (xv + yw)∂v − (xw − yv)∂w + v(yv − xw)∂z } . The following relations hold [U, X1 ] = −2X2 , [U, X2 ] = 2X1 (where X2 = JX1 ) which means that [U, Γ(H)] ⊂ Γ(H). Moreover, since du = 21 (dx ∧ dy + dv ∧ dw) we get U ydu = 0. We have also [X1 , X2 ] = −16(v 2 + w2 ) U − 8(v 2 + w2 ) ξ. ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 15 Now we are able to write the expression of the connection ∇. We have  ∇X1 X1 = −4xX1 + 4yX2 , ∇X2 X2 = 4xX1 − 4yX2      ∇X1 X2 = −4yX1 − 4xX2 − 8(v 2 + w2 ) U − 4(v 2 + w2 ) ξ      ∇U X1 = −2X2 , ∇U X2 = 2X1 . Computing the curvature tensor of ∇ we obtain RX1 X2 X1 = −64(v 2 + w2 )X2 , RX1 X2 X2 = 64(v 2 + w2 )X1 other components being zero. It follows ρ(R)(X1 , X1 ) = ρ(R)(X2 , X2 ) = 256(v 2 + w2 ) and ρ(R)(X1 , X2 ) = 0. Example 3. Consider now the following hypersurface M in R5 M = {(x, y, z, v, w) ∈ R5 : w = xy} . The tangent space of M is spanned by  U = µ2 (−x∂x + y∂y + ∂v ) , ξ = 2∂z  X1 = ∂x + x∂v + y∂w + y(1 + x2 ) ∂z , X2 = ∂y − y∂v + x∂w − xy 2 ∂z . In order to verify if U ydu = 0 or not we compute [U, X1 ] (since X1 ∈ Γ(H)). We get [U, X1 ] = − x 3 +3x+xy 2 µ2 U+ 2(1−x2 +y 2 ) µ3 X1 + 4xy µ3 X2 so, u([U, X1 ]) 6= 0, which means that U ydu 6= 0. This example proves that the condition (2.17) is not automatically satisfied. The next example is inspired from the following theorem ([7], Th. 5.2, p. 185): Let M be a compact orientable pseudo-umbilical hypersurface of S 2n+1 (n ≥ 2). Then M is S 2n−1 (r1 ) × S 1 (r2 ) , r12 + r22 = 1. Example 4. (PUH) Let M = S 3 (r1 ) × S 1 (r2 ) with r12 + r22 = 1 be a pseudoumbilical hypersurface in S 5 (⊂ R6 ) as a Sasakian space form. On R6 consider ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 16 global coordinates x, y, v, w, s, t so, on M we have |p1 | = r1 and |p2 | = r2 where p1 = (x, y, v, w), p2 = (s, t) ans | · | denotes the usual Euclidean norm. Consider ξ1 = 1 1 1 (−y, x, −w, v) , X1 = (−v, w, x, −y) , X2 = (−w, −v, y, x) r1 r1 r1 which form an orthonormal frame on S 3 (r1 ) and ξ2 = r12 (−s, t) ∈ χ(S 1 (r2 )). Consider also the following contact forms on S 3 (r1 ) and S 1 (r2 ) respectively η1 = 1 1 (−y dx + x dy − w dv + v dw) , η2 = (−s dt + t ds). r1 r2 With these notations, the (almost) contact structure on S 5 is given by ξ = r1 ξ1 + r2 ξ2 , η = r1 η1 + r2 η2 , φX1 = X2 , φX2 = −X1 . The unit normal vector field on M is N = − rr21 p1 + rr21 p2 thus U = −r2 ξ1 + r1 ξ2 . We have obtained a global frame on M satisfying  2r2  [X1 , ξ] = 2X2 , [X1 , U ] = − r1 X2 2 [X2 , ξ] = −2X1 , [X2 , U ] = 2r r 1 X1  2r2 [X1 , X2 ] = −2ξ + r1 U. Moreover, on M we have du = − rr21 dη. Computing the torsion free adapted connection we obtain: r2 ∇X1 X1 = 0 , ∇X1 X2 = −ξ + U , ∇X2 X2 = 0 r1 the other expressions are easy deducible from the relations above. The non-vanishing components of the curvature tensor are RX 1 X 2 X 1 = − 4 4 X2 , RX1 X2 X2 = 2 X1 r12 r1 and hence ρ(R)(X1 , X1 ) = ρ(R)(X2 , X2 ) = 4 , ρ(R)(X1 , X2 ) = 0. r12 Consequently we have a CR-Einstein manifold which is never flat. Example 5. The condition (2.17) U ydu = 0 is a quite strong condition which in general yields to a PDE system. Even that the dimension of the ambient manifold is small, this PDE system is rather complicate. Yet, if the submanifold M in R5 is defined by a function depending only on one or two variables, the condition (2.17) ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ 17 is automatically satisfied. Let us, for example, consider f (x, y, v, w) = y 2 + w2 − r2 (i.e. M = S 1 × R3 ). On M we have: N = 1r (yB1 + wB2 ) - the unitary normal vector field; U = − 1r (yA1 + wA2 ); ¸ X1 = 1r (wA1 − yA2 ) -unitary, orthogonal and belonging to H(M ) X2 = 1r (wB1 − yB2 ) where A1 = 2 (∂x + y∂z ), B1 = 2∂y , A2 = 2 (∂v + w∂z ), B2 = 2∂w (see for more details [1], [2]). An easy computation gives the expression of Weingarten operator namely, AX1 = 0, AX2 = − 2r X2 , Aξ = −U and AU = −ξ so M does not belong to the case (λ − µ) described in the previous section. Computing the torsion free adapted connection we obtain ∇X1 X1 = 0 , ∇X2 X2 = 0 , ∇X1 X2 = −ξ − ∇U X 1 = 0 , ∇U X2 = 1 U r 2 X1 , ∇ξ X1 = − , ∇ξ X2 = 0 . r Consequently ρ(R)(X1 , X1 ) = 0, ρ(R)(X1 , X2 ) = 0, ρ(R)(X2 , X2 ) = 4 r2 and hence M is not CR-Einstein. Example 6. Looking at the examples 3 and 5 we are interested to study a submanifold in R5 defined by a function f depending on x, y and w (with fw 6= 0) and having the property (2.17). In this case one gets the following PDE’s system   −fw fy fxy + fw fx fyy + fy2 fxw − fx fy fyw = 0  −fw fx fxy + fw fy fxx + fx2 fyw − fx fy fxw = 0 . Since fw = 6 0 let us consider that the submanifold M is given explicitely by w = r(x, y), r ∈ C ∞ (D), D ⊂ R2 . Then we obtain the following PDE’s system involving r:   rx ryy − ry rxy = 0  rx rxy − ry rxx = 0 having the solution r(x, y) = r(y+ρx) with an arbitrary function r ∈ C ∞ (I),I ⊂ R, r′ 6= 0 and ρ ∈ R. On M we have: µ2 = 1 + (ρ2 + 1)r′2 , ⋄ Marian Ioan Munteanu ⋄ CR-structures of CR-codimension 2 ⋄ N= U= X1 = X2 = 1 ′ ′ µ (−ρ r A1 − r B1 + B2 ) - the unitary normal vector field, 1 ′ ′ µ (r A1 − ρ r B1 − A2 ), ¸ 1 ′ ′ µ (A1 + r A2 + ρ r B2 ) - unitary, orthogonal and belonging 1 ′ ′ µ (B1 − ρ r A2 + r B2 ) 18 to H(M ). After some easy computations, we get the expression of the Weingarten operator: ′′ ′′ Aξ = −U , AU = −ξ, AX1 = 2ρµ3r (ρX1 + X2 ) and AX2 = 2r µ3 (ρX1 + X2 ), so, M is TCG if and only if M is a hyperplane (r′′ = 0) and if and only if M is minimal ′′ 2 +1) N ). (the mean curvature is H = r (ρ 2µ3 Computing the adapted connection we obtain ∇X1 X2 = −∇X2 X1 = −ξ + ′′ (ρ2 +1)r′′ µ3 U (X1 − ρX2 ) ∇U X1 = ρ ∇U X2 = − 2ρr µ3 all other combinations being zero. Consequently, the adapted connection is flat if and only if M is a hyperplane. References [1] D.E.Blair, Contact manifolds in Riemannian Geometry, Lect. Notes in Math., 509, Springer-Verlag, 1976. [2] D.E.Blair, Riemannian geometry of contact and symplectic manifolds, Progress in Mathematics, 203, Birkhäuser, 2001. [3] S.I.Goldberg & K.Yano, On normal globally f -manifold, Tôhoku Math. J., 22 (1970), 362-370. [4] A.Lotta & A.M.Pastore, The Tanaka-Webster connection for almost S-manifolds and Cartan geometry, Archivum Mathematicum (Brno), 40 (2004), 47-61. [5] P.Matzeu & V.Oproiu, The Bochner type curvature tensor of pseudoconvex CR structures, SUT J. of Math., 31, 1 (1995), 1-16. [6] P.Matzeu & V.Oproiu, The Bochner type curvature tensor of pseudo-convex CR structures on real hypersurfaces in complex space forms, J. of Geometry, 63, (1998), 134146. [7] K.Yano & M.Kon, CR-submanifolds of Kählerian and Sasakian manifolds, Progress in Mathematics, 30, Birkhäuser, Boston, Basel Stuttgart, 1983. University ’Al.I.Cuza’ of Iaşi, Faculty of Mathematics, Bd. Carol I, no.11, 700506 Iaşi Romania e-mail: [email protected], marian ioan [email protected] Received, November, 2004