SOME REMARKS ON INVARIANT SURFACES AND

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We give a general for- mula for computing the extrinsic curvature of an invariant surface. We classify the integrable. Killing vector fields on BCV-spaces and we ...
SOME REMARKS ON INVARIANT SURFACES AND THEIR EXTRINSIC CURVATURE F. MERCURI, S. MONTALDO, AND I. I. ONNIS

Abstract. We introduce the notion of integrable invariant surfaces as those which are invariant under the action of a one-parameter subgroup of isometries generated by an integrable Killing vector field, that is with integrable horizontal distribution. We give a general formula for computing the extrinsic curvature of an invariant surface. We classify the integrable Killing vector fields on BCV-spaces and we prove that all Hopf-cylinders in these spaces have constant extrinsic curvature.

An important geometric class of surfaces in a three dimensional manifold is that of invariant surfaces, that is, as described in the next section, surfaces which are invariant under the action of a one-parameter subgroup of isometries of the ambient space. Invariant surfaces have been classified, according to the value of their Gaussian or mean curvature, in many remarkable three dimensional spaces (see, for example, [CPR1, CPR2, FMP, MO1, MO2, On, To]). More recently some authors have considered the problem of classifying surfaces, in a three dimensional space, accordingly to the value of their extrinsic curvature. The case of complete surfaces with constant positive extrinsic curvature in the product space M 2 × R has been considered by Espinar, G´ alvez and Rosenberg in [EG]. In [Lo] L´opez studied invariant surfaces in the homogenous space Sol with constant extrinsic curvature. In this paper we give a general formula for computing the extrinsic curvature Ke of an invariant surface in a three dimensional manifold. This formula depends on the profile curve that generates the invariant surface and on the geodesic torsion of the orbits of the one-parameter subgroup of isometries. Moreover, this formula suggests to study the surfaces which are invariant under the action of a one-parameter subgroup of isometries with torsion-free orbits. We will show that this is the case when the Killing vector field, associated with the one-parameter group of isometries, has integrable horizontal distribution. Motivated by this property we shall call: a Killing vector field integrable when its horizontal distribution is integrable; an invariant surface integrable when it is invariant under the action of a one-parameter subgroup of isometries generated by an integrable Killing vector field. In the last part we consider the Bianchi-Cartan-Vranceanu spaces and we classify the integrable Killing vector fields on these spaces. Finally, we prove that all Hopf-cylinders in Bianchi-CartanVranceanu spaces have constant extrinsic curvature. 1. Killing vector fields and invariant surfaces Let (N 3 , g) be a three dimensional Riemannian manifold and let X be a Killing vector field on N . Then X generates a one-parameter subgroup GX of the isometry group. If N/GX is connected, from the Principal Orbit Theorem ([Pa]), the principal orbits are all diffeomorphic and the regular set Nr , consisting of points belonging to principal orbits, is open and dense in N . Moreover, the quotient space Nr /GX is a connected differentiable manifold and the 2010 Mathematics Subject Classification. 53C42, 53B15. Key words and phrases. Invariant surfaces, Killing vector fields, extrinsic curvature. The first author was supported by: CNPq - Brazil. The second author was supported by: grant for the startup of young researchers, University of Cagliari - Italy. The third author was supported by: visiting professors program, Regione Autonoma della Sardegna - Italy and CNPq - Brazil. 1

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F. MERCURI, S. MONTALDO, AND I. I. ONNIS

quotient map π : Nr → Nr /GX is a submersion. It is well known (see, for example, [Ol]) that Nr /GX can be locally parametrized by the invariant functions of the Killing vector field X. If {ξ1 , ξ2 } is a complete set of invariant functions on a GX -invariant subset of Nr , then we define P2 ij ij the quotient metric by g˜ = i,j=1 h dξi ⊗ dξj where (h ) is the inverse of the matrix (hij ) with entries hij = g(∇ξi , ∇ξj ). If we equip the quotient space Nr /GX with the quotient metric, then the quotient map π : Nr → Nr /GX becomes a Riemannian submersion. We shall denote by V = ker(dπ) the vertical distribution and by H = V ⊥ the horizontal ones. By construction, the vertical distribution at a point x ∈ Nr is the tangent space at x to the orbit of the point x under the action of GX . Thus the vertical distribution is always integrable while we shall call a Killing vector field integrable if the horizontal distribution, restricted to Nr , is integrable. Let now f : M 2 → (N 3 , g) be an immersion from a surface M 2 into N 3 and assume that f (M ) ⊂ Nr . We say that f is a GX -equivariant immersion, and f (M ) a GX -invariant surface of N , if there exists an action of GX on M 2 such that for any x ∈ M 2 and g ∈ GX we have f (gx) = gf (x). A GX -equivariant immersion f : M 2 → (N 3 , g) induces an immersion f˜ : M/GX → Nr /GX between the orbit spaces as shown in the following diagram f

(M 2 , gf ) −−−−→   y

(N 3 , g)   πy



M 2 /GX −−−−→ (Nr3 /GX , g˜) where π is a Riemannian submersion and with gf = f ∗ (g) we have denoted the pull-back metric. We now give a local description of the GX -invariant surfaces of N 3 . Let γ˜ : (a, b) ⊂ R → (N 3 /GX , g˜) be a curve parametrized by arc length and let γ : (a, b) ⊂ R → N 3 be a lift of γ˜ , such that dπ(γ 0 ) = γ˜ 0 . If we denote by φr , r ∈ (−, ), the local flow of the Killing vector field X, then the map (1.1)

ψ : (a, b) × (−, ) → N 3 ,

ψ(t, r) = φr (γ(t)),

defines a parametrized GX -invariant surface. Conversely, if f (M 2 ) is a GX -invariant immersed surface in N 3 , then f˜ defines a curve in (N 3 /GX , g˜) that can be locally parametrized by arc length. The curve γ˜ is generally called the profile curve. 2. Extrinsic curvature of an invariant surface Let f : M 2 → (N 3 , g) be a GX -invariant surface, then, locally, the surface f (M 2 ) can be parametrized, using (1.1), by ψ(t, r) = φr (γ(t)). The pull-back metric can be written as gf = Edt2 + 2F dtdr + Gdr2 where  0 0  E = g(ψt , ψt ) = g(dφ(γ ), dφ(γ )) 0 F = g(ψt , ψr ) = g(dφ(γ ), X)   G = g(ψr , ψr ) = g(X, X) = ω 2 . Since the r-coordinate curves are the orbits of the action of the one-parameter group of isometries GX , the coefficients of the metric do not depend on r. As γ is the lift of γ˜ , with respect to the Riemannian submersion π, we have that dπ(ψt ) = γ˜ 0 and dπ(ψr ) = 0. Let e be a local unit vector field tangent to the surface and horizontal with respect to π, i.e., e ∈ H. Then, since dπ(ψt ) = γ˜ 0 has norm 1 and π is a Riemannian submersion, ψt can be decomposed as ψt = g(ψt , X)

F X + e = X + e. g(X, X) G

Calculating the norm yields to EG − F 2 = G = ω 2 .

SOME REMARKS ON INVARIANT SURFACES AND THEIR EXTRINSIC CURVATURE

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Let η be a local unit vector field normal to f (M ), as π is a Riemannian submersion, we have that 0 = g(η, e) = g˜(dπ(η), dπ(e)) = g˜(dπ(η), γ˜ 0 ) and ||dπ(η)|| = ||η|| = 1. So we can choose η so that dπ(η) = J(e γ 0 ), where J is the complex structure on N/GX given by counter-clockwise rotation of π/2. To compute the extrinsic curvature of the immersion f : M 2 → (N 3 , g) we calculate the coefficients of the second fundamental form. We obtain 2F F2 g(∇X e, η) + g(∇e e, η); h11 = g(∇ψt ψt , η) = 2 g(∇X X, η) + G G F h12 = g(∇X ψt , η) = g(∇X X, η) + g(∇X e, η); G 1 1 h22 = g(∇X X, η) = − η g(X, X) = − η(ω 2 ) = −ω 2 η(ln ω). 2 2 Therefore h11 h22 − h212 = g(∇e e, η) N − g(∇X e, η)2 . As X is a Killing vector field and e is horizontal, we have that [e, X] = 0. Therefore, from g(e, X) = 0, it results that g(∇e e, X) = −g(∇e X, e) = −g(∇X e, e) = 0. So ∇e e is horizontal. Consequently, ˜ γ˜ 0 γ˜ 0 , J(e g(∇e e, η) = g˜(dπ(∇e e), dπ(η)) = g˜(∇ γ 0 )) = κ2 , where with κ2 we have denoted the curvature with sign of the profile curve. To compute g(∇X e, η) we recall that the geodesic torsion of a curve α of N that lies on the surface M ⊂ N is defined by g(J(α0 ), Aη (α0 )) , τ (α) = kα0 k2 where J is the complex structure on M given by counter-clockwise rotation of π/2 and Aη denotes the shape operator of M in N . Now, using that J(X) = ±e ||X|| = ±e ω, it results that g(∇X e, η)|(t,r)

= −g(e, ∇X η)|(t,r)

g(J(X), Sη (X)) |(t,r) = ∓ω τ (t, r), ω where τ (t, r) is the geodesic torsion of the orbit of X through γ(t) calculated at φr (γ(t)). A direct computation shows that the orbits of a Killing vector field have constant geodesic torsion, thus, in (2.1), τ is a function only of t. Consequently we have = ∓

(2.1)

Proposition 2.1. Let f : M 2 → (N 3 , g) be a GX -invariant surface locally parametrized by (1.1). Then the extrinsic curvature of the immersion depends only on the profile curve and it is given by h11 h22 − h212 = −κ2 η(ln ω) − τ 2 , EG − F 2 where ω = kXk, κ2 is the curvature with sign of the profile curve and τ is the geodesic torsion of the orbits. (2.2)

Ke =

The geodesic torsion of the orbits has the following interpretation Proposition 2.2. Let f : M 2 → (N 3 , g) be a GX -invariant surface. Then the geodesic torsion of the orbits of X is zero if an only if the horizontal distribution H is integrable.

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F. MERCURI, S. MONTALDO, AND I. I. ONNIS

Proof. From (2.1), τ = 0 if and only if g(∇X e, η) = 0. Using the O’Neill tensor A and that ∇e η = A(e, η) + (∇e η)H , we have g(∇e X, η) = −g(X, ∇e η) 1 = −g(X, A(e, η)) = − g(X, [e, η]V ). 2 Since the vertical space is 1-dimensional and X is not zero on M , we conclude that τ = 0 if and only if [e, η] ∈ H.  g(∇X e, η)

=

Using the above proposition there exists two classes of invariant surfaces in a 3-dimensional manifold, depending on the integrability of the Killing vector field X. We shall call a surface an invariant integrable surface if it is invariant under the action of a one-parameter subgroup of isometries generated by an integrable Killing vector field. 3. The Bianchi-Cartan-Vranceanu spaces (BCV-spaces) The Bianchi-Cartan-Vranceanu space M 3 (`, m), `, m ∈ R, (see [Bi, Ca, Vra]) is given by the set {(x, y, z) ∈ R3 : 1 + m (x2 + y 2 ) > 0} equipped with the Riemannian metric (3.1)

g`,m =

 2 ` ydx − xdy dx2 + dy 2 + dz + , F2 2 F

where F = 1 + m(x2 + y 2 ). The geometric interest of these spaces lies in the following fact: the family of metrics (3.1) includes all three-dimensional homogeneous metrics whose group of isometries has dimension 4 or 6, except for those of constant negative sectional curvature. The Riemannian manifold (M 3 (`, m), g`,m ) can be described, according to the values of m and `, as follows (see, for example, [Ko]): • if ` = m = 0, is the flat R3 ; • if ` = 0 and m 6= 0, is the product of a surface with constant Gaussian curvature 4m and the real line R; • if ` 6= 0 and 4m − `2 = 0, has non negative constant sectional curvature; • if ` 6= 0, m > 0 and 4m − `2 6= 0, is locally SU (2); f • if ` 6= 0 and m < 0, is locally SL(2, R); • if ` 6= 0 and m = 0, is the Heisenberg space H3 . With respect to the metric g`,m , an orthonormal frame is (3.2)

E1 = F

∂ `y ∂ − , ∂x 2 ∂z

E2 = F

∂ `x ∂ + , ∂y 2 ∂z

E3 =

∂ , ∂z

and the Lie algebra of the infinitesimal isometries of M 3 (`, m) admits the following basis of Killing vector fields (see, for example, [Pi]):

(3.3)

h 2my 2 i 2mxy `y X1 = 1 − E1 + E2 + E3 , F F F h 2mxy 2mx2 i `x X2 = E1 + 1 − E2 − E3 , F F F y x ` (x2 + y 2 ) X3 = − E1 + E2 − E3 , F F 2F ∂ X4 = . ∂z

SOME REMARKS ON INVARIANT SURFACES AND THEIR EXTRINSIC CURVATURE

For later use, we recall that, rewriting the metric g`,m =

P3

i=1

5

θi ⊗ θi , where

dx dy ` θ1 = , θ2 = , θ3 = dz + (y θ1 − x θ2 ), F F 2 P3 we have that dθi = j=1 θj ∧ θji , i = 1, 2, 3, with

(3.4)

θ12 = 2my θ1 − 2mx θ2 −

` 3 θ = −θ21 , 2

` θ13 = − θ2 = −θ31 , 2 ` θ23 = θ1 = −θ32 . 2 3.1. Integrable Killing vector fields. P4In this section we shall find when a Killing vector field of BCV-spaces is integrable. Let X = i=1 ai Xi , ai ∈ R, be a Killing vector field of M 3 (`, m). Then X can be decomposed, with respect to the frame (3.2), as (3.5)

X=

3 X

λi Ei ,

i=1

where  2mxy y 2my 2  a1 + a2 − a3 , λ1 = 1 − F F F  2mxy 2mx2  x λ2 = a1 + 1 − a2 + a3 , F F F `y `x ` (x2 + y 2 ) λ3 = a1 − a2 − a3 + a4 . F F 2F

(3.6)

Let ρ =

P3

i=1

λi θi be the 1-form dual to X. Taking into account that dλi = F [(λi )x θ1 + (λi )y θ2 )], 1

1

i = 1, 2, 3,

2

dθ = 2my θ ∧ θ , dθ2 = −2mx θ1 ∧ θ2 , dθ3 = −` θ1 ∧ θ2 , it results that ρ ∧ dρ =λ1 θ1 ∧ dλ3 ∧ θ3 + λ2 θ2 ∧ dλ3 ∧ θ3 + λ3 (θ3 ∧ dλ1 ∧ θ1 + θ3 ∧ dλ2 ∧ θ2 ) (3.7)

+ λ3 (λ1 θ3 ∧ dθ1 + λ2 θ3 ∧ dθ2 + λ3 θ3 ∧ dθ3 )    = F λ3 (λ2 )x − λ2 (λ3 )x + λ1 (λ3 )y − λ3 (λ1 )y + 2mλ3 (y λ1 − x λ2 ) − ` λ23 θ1 ∧ θ2 ∧ θ3 =A`,m (x, y) θ1 ∧ θ2 ∧ θ3 .

From Frobenius’s Theorem X is integrable if and only if ρ∧dρ = 0. Thus we study the equation (3.8)

A`,m (x, y) = 0,

for all the values of ` and m. We have Theorem P4 3.1. According to the values of ` and m the integrability of a Killing vector field X = i=1 ai Xi is given by the following: (a) when ` = m = 0 if a3 a4 = 0;

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F. MERCURI, S. MONTALDO, AND I. I. ONNIS

(b) when ` = 0 and m 6= 0 if either a1 = a2 = a3 = 0 or a4 = 0; (c) when ` 6= 0 and 4m − `2 = 0 if (a21 + a22 − a24 ) ` + 2a3 a4 = 0; (d) when ` 6= 0 and 4m − `2 6= 0 never. Proof. First A`,m (x, y) can be written as A`,m (x, y) = −

4 1 X Ai (y) xi , 4F 2 i=0

where

(3.9)

 A4 (y) =` (`2 − 4m) a23 − 4m (`2 − 2m) a3 a4 + 4` m2 (a24 − a21 − a22 ),      A3 (y) =4(`2 − 4m) a2 (` a3 − 2m a4 ),       A2 (y) =2 A4 (y) y 2 − 4(`2 − 4m) a1 (` a3 − 2m a4 ) y      + 4` [a22 (`2 − 4m) + 2m (a24 − a21 − a22 ) − ` a3 a4 ],  A1 (y) =4(`2 − 4m) a2 [(` a3 − 2m a4 ) y 2 − 2` a1 y − 2a4 ],       A0 (y) =A4 (y) y 4 − 4(`2 − 4m) a1 (` a3 − 2m a4 ) y 3      + 4` [a21 (`2 − 4m) + 2m (a24 − a21 − a22 ) − ` a3 a4 ] y 2     + 8(`2 − 4m) a1 a4 y − 4 [2a3 a4 + ` (a21 + a22 − a24 )],

Since A`,m (x, y) is zero in the regular part of the action of GX , which is dense in M , it follows that Ai (y) = 0, i = 0, . . . , 4. Now, replacing the values of ` and m in (3.9) the case (a), (b) and (c) are immediate. We prove (d). First observe that X3 is the only Killing vector field that is null at the origin. In this case (3.8) is equivalent to `( `2 − 4m) (x2 + y 2 )2 = 0. Therefore X3 is integrable if and only if ` = 0 or `2 = 4m. Thus we can assume that X is not null at the origin. From A2 (y) = 0 we have (3.10)

a22 (`2 − 4m) + 2m (a24 − a21 − a22 ) − ` a3 a4 = 0

while A0 (y) = 0 implies that (3.11)

a21 (`2 − 4m) + 2m (a24 − a21 − a22 ) − ` a3 a4 = 0

and (3.12)

2a3 a4 + ` (a21 + a22 − a24 ) = 0.

From (3.10) and (3.11), it follows that (3.13)

|a1 | = |a2 |.

Also, from A1 (y) = 0, we have two possibilities: (i) either (` a3 − 2m a4 ) y 2 − 2` a1 y − 2a4 = 0, for all y, (ii) or a2 = 0.

SOME REMARKS ON INVARIANT SURFACES AND THEIR EXTRINSIC CURVATURE

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If (i) occurs, then a1 = a3 = a4 = 0 and from (3.12) a2 = 0. In the case (ii) holds, from (3.13), a1 = 0 and (3.11) reduces to a4 (2m a4 − ` a3 ) = 0. If a4 = 0, from A4 (y) = 0 it follows that a3 = 0. Otherwise, if a3 = 2m a4 /`, from (3.12) we have (4m − `2 ) a24 = 0, i.e. a4 = 0. Therefore a3 = 0 and we conclude.  3.2. Hopf-cylinders in BCV-spaces. A special class of invariant surfaces in BCV-spaces is that of surfaces invariant under the action of the one parameter subgroup of isometries generated by X4 , that is by vertical translations. In this case the orbit space is M 2 (4m) = {(x, y) ∈ R2 : 1 + m (x2 + y 2 ) > 0}, that, endowed with the orbital metric g˜ = (dx2 + dy 2 )/F 2 , is of constant Gaussian curvature 4m. The projection π : M 3 (`, m) → M 2 (4m), π(x, y, z) = (x, y), is a Riemannian submersion which is often called the Hopf-fibration. The invariant surfaces are, in this case, the inverse image of curves in M 2 (4m) under π and are called Hopf-cylinders. Theorem 3.2. Let γ : I → M 2 (4m) be a curve parametrized by arc length and S = π −1 (γ(I)) the Hopf-cylinder in M 3 (`, m). Then the extrinsic curvature of S is constant and equal to −`2 /4. Proof. At a point p ∈ M 3 (`, m) the vertical space of the submersion π is Vp = ker(dπp ) = span(E3 ) and the horizontal space Hp = span(E1 , E2 ). As the curve γ(t) = (x(t), y(t)) is parametrized by arc length, we have that g˜(γ 0 , γ 0 ) =

(3.14)

x0 (t)2 + y 0 (t)2 = 1, F (t)2

where F (t) = 1 + m (x(t)2 + y(t)2 ). Consequently the vector fields e1 =

x0 (t) y 0 (t) E1 + E2 F (t) F (t)

and

e2 = E3

give an orthonormal frame tangent to S and η=−

y 0 (t) x0 (t) E1 + E2 F (t) F (t)

is a unit normal vector field of S. Observe that as [e1 , e2 ] = 0, i.e. ∇e1 e2 = ∇e2 e1 , from g`,m (e1 , e2 ) = 0 it results that 0 = g`,m (∇e1 e1 , e2 ) + g`,m (e1 , ∇e1 e2 ) = g`,m (∇e1 e1 , e2 ) + g`,m (e1 , ∇e2 e1 ) = g`,m (∇e1 e1 , e2 ) = g`,m (∇e1 e1 , E3 ). Therefore ∇e1 e1 is horizontal. In addition dπ(e1 ) = γ 0 and dπ(η) = J(γ 0 ), so h11 = g`,m (∇e1 e1 , η) = gm (dπ(∇e1 e1 ), dπ(η)) = gm (∇γ 0 γ 0 , J(γ 0 )) = κ2 , where κ2 is the curvature with sign of γ. Also, as ` ∇E3 E1 = − E2 , 2

∇E3 E2 =

we obtain that h12 = g`,m (∇e2 e1 , η) = g`,m

` E1 , 2

∇E3 E3 = 0,

 ` x0 ` ` y0 E1 − E2 , η = − 2F 2F 2

and h22 = g`,m (∇e2 e2 , η) = 0.

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F. MERCURI, S. MONTALDO, AND I. I. ONNIS

In summary, the matrix of the second fundamental form of S with respect to the frame {e1 , e2 } is   κg − 2`   ` −2 0 and the extrinsic curvature of S is −`2 /4.  References [Bi] L. Bianchi. Gruppi continui e finiti. Ed. Zanichelli, Bologna, 1928. ´ Cartan, Le¸cons sur la g´ [Ca] E. eom´ etrie des espaces de Riemann, Gauthier Villars, Paris, 1946. [CPR1] R. Caddeo, P. Piu and A. Ratto, SO(2)-invariant minimal and constant mean curvature surfaces in three dimensional homogeneous spaces, Manuscripta Math. 87 (1995), 1–12. [CPR2] R. Caddeo, P. Piu and A. Ratto, Rotational surfaces in H3 with constant Gauss curvature, Boll. Un. Mat. Ital. B 10 (1996), 341–357. [EG] J. M. Espinar, J. A. G´ alvez and H. Rosenberg, Complete surfaces with positive extrinsic curvature in product spaces, Comment. Math. Helv. 84 (2009) 351–386. [FMP] C. B. Figueroa, F. Mercuri and R. H. L. Pedrosa, Invariant surfaces of the Heisenberg groups, Ann. Mat. Pura Appl. 177 (1999), 173–194. [Ko] O. Kowalski, Spaces with volume-preserving symmetries and related classes of Riemannian manifolds, Rend. Sem. Mat. Torino (1983), 131–159. [Lo] R. Lopez, Invariant surfaces in homogenous space Sol with constant curvature, arXiv:0909.2550. [MO1] S. Montaldo and I. I. Onnis, Invariant CMC surfaces in H2 × R, Glasg. Math. J. 46 (2004), 311–321. [MO2] S. Montaldo and I. I. Onnis, Invariant surfaces in a three-manifold with constant Gaussian curvature, J. Geom. Phys. 55 (2005), 440–449. [Ol] P. J. Olver, Application of Lie groups to differential equations, GTM 107, Springer-Verlag, New York (1986). [On] I.I. Onnis, Invariant surfaces with constant mean curvature in H2 × R, Ann. Mat. Pura Appl. 187 (2008), 667–682. [Pa] R. S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. 73 (1961), 295-323. [Pi] P. Piu, Sur certains types de distributions non-int´ egrables totalement g´ eod´ esiques, Th` ese de Doctorat, Universit´ e de Haute-Alsace, Mulhouse, 1988. [To] P. Tomter, Constant mean curvature surfaces in the Heisenberg group, Proc. Sympos. Pure Math. 54, 485–495, Amer. Math. Soc., Providence, RI, 1993. [Vra] G. Vranceanu, Le¸cons de g´ eom´ etrie diff´ erentielle, Ed. Acad. Rep. Pop. Roum., vol I, Bucarest, 1957. Departamento de Matematica, C.P. 6065, IMECC, UNICAMP, 13081-970, Campinas, SP, Brazil E-mail address: [email protected] ` degli Studi di Cagliari, Dipartimento di Matematica, Via Ospedale 72, 09124 Cagliari Universita E-mail address: [email protected] ´ tica, C.P. 668, ICMC, USP, 13560-970, Sa ˜ o Carlos, SP, Brasil Departamento de Matema E-mail address: [email protected]