On Closed Concircular Almost Contact Riemannian Manifolds

Fernando Etayo, Radu Rosca and Rafael Santamaría

Abstract

A class of almost contact Riemannian manifolds (ACRM) (which we call closed concircular ACRM) is defined and studied, showing that such a manifold is a local product and obtaining some important harmonic properties.

1991 Mathematics Subject Classification: 53C15, 53B20, 53C55. Key words: almost contact manifold, concircular and ϕ-soldering- vector fields, isoparametric function.

Introduction

Almost contact structures are, in some sense, the odd-dimensional analogous to almost complex structures. In fact, if M is an almost contact manifold, it is easily seen that M×R become an almost complex manifold and this property has been used by Oubiña [?] to obtain a classification of almost contact metric manifolds. Moreover, almost contact and almost complex structures are the simplest examples of a f-structure (see e.g. [?]).

Additional structures on an almost contact manifold have been introduced in the last decades in several papers (see e.g [?]). In our work we study the geometry of an almost contact Riemannian manifold when its Reeb vector field ξ satisfy two properties: it is concircular and U^{♭} is closed, where U = ∇_{ξ}ξ.

Now we intoduce some notation and explain the main results of the paper. Let (M, ϕ, η, ξ, g) be almost contact Riemannian manifold, where the structure tensors (ϕ, Ω, η, ξ) mean the tensor field, the 2-form, the 1-form and its dual η^{♯} = ξ, respectively. One has

\begin{matrix}ϕ^{2} = −Id + η ⊗ ξ, η ∧ Ω^{m} ≠ 0; \\ g(ϕX, ϕY) = g(X, Y) − η(X)η(Y); \end{matrix}

for any vector fields X, Y on M, where Ω(X, Y) = g(ϕX, Y).

We assume in this paper that ξ is U-concircular, i.e.,

∇ξ = η ⊗ U

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 75-88 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

where U ∈ Kerη will be called the generative of ξ and that U^{♭} is a closed 1-form. In this case we find

dΩ = 0, dη = U^{♭} ∧ η ≠ 0, η ∧ dη = 0,

which implies that M is not a cosymplectic nor a contact manifold.

In section 2 we prove important properties of η, Ω and U. In the last part of this section we assume that the manifold satisfies a technical condition and we prove, among other properties, the following structure equations

(i) L_{U}η = ∥U∥^{2}η; L_{U}Ω = 2λΩ; λ = const. i.e., U defines an infinitesimal conformal transformation of the almost contact Riemannian structure defined on M.

(ii) Any such M , may be viewed as the locally product of the integral submanifolds of D_{U} = Span{U, ξ} and D_{U}^{⊥}.

(iii) The transversal curvature forms Θ_{0}^{A} ( A ∈ {1, …, 2m}) are monomial and they define a closed vector valued 2-form

Π = ∑ Θ_{0}^{A} ⊗ e_{A} = λη ∧ I + (U^{♭} ∧ η) ⊗ U.

In section 3 one afirms that U^{♭} and η are eigenfunctions of Δ and if

η_{q} = η ∧ Ω^{q},

then the Lie derivatives with respect to ξ are expressed by

L_{ξ}η_{q} = −U^{♭} ∧ Ω^{q}.

In the last section a vector field X such that

∇X = fϕdp − g(X, U)η ⊗ ξ

is defined as a ϕ-soldering vector field (abr. ϕs). If M carries such a vector field, then Ω is invariant by X, and it moves to a potential form and M is foliated by 3-dimensional submanifolds M_{X} tangent to Span{ϕX, ξ, U}.

It is also proved that ϕX defines an infinitesimal homothety of Ω and both X and ϕX define infinitesimal conformal transformations of ξ.

1 Preliminaries and Notations

First, we shall explain the notation used in the paper.

Let (M, g) be a n-dimensional connected Riemannian manifold and let ∇ be the covariant differential operator defined by the metric tensor g (we assume that M is oriented and ∇ is the Levi-Civita connection).

We denote by F(M) the ring of real functions, T_{q}^{p}(M) the F(M)-module of (p, q) tensor fields, T M (resp T^{∗}M) the tangent (resp. cotangent) bundle of M and ⋀^{r}(M) the F(M)-module of r-forms, with ⋀ M = ⊕_{r=0}^{n} ⋀^{r}(M), X M the set of sections of the tangent bundle T M and

♭ : T M → T^{∗}M, ♯ : T^{∗}M → T M

the musical isomorphisms defined by g.

If X_{i} ∈ X M are vector fields, Span{X_{1}, …, X_{k}} is the distribution spanned by them.

We denote by A^{q}(M, T M) = ΓHom(⋀^{q} T M, T M) the set of vector valued qforms, and following [?] we write for the covariant derivative with respect to ∇, d^{∇} : A^{q}(M, T M) → A^{q+1}(M, T M). The vector valued 1-form dp ∈ A^{1}(M, T M) is the identity vector valued 1-form and is called the soldering form of M [?]. As is well known, A^{1}(M, T M) is isomorphic to T_{1}^{1}, and the soldering form dp corresponds with the identity (1, 1) tensor field, which will be denoted as I. If ϕ ∈ T_{1}^{1} then ϕdp ∈ T_{1}^{1} and one has the left inner product < X, ϕdp >= ϕX =< ϕX, dp >. As ∇ is a symmetric connection one obtains d^{∇}(dp) = 0.

We write for T (resp. R) the torsion (resp. curvature) tensor of ∇. Then, for any vector field Z on M , the second covariant differential is defined as ∇^{2}Z = d^{∇}(∇Z) and satisfies

∇^{2}Z(V, W) = R(V, W)Z; V, W ∈ X M (1.1)

The operator d^{ω} = d+e(ω) acting on ΛM, where e(ω) means the exterior product by the closed 1-form ω is called the cohomology operator [?]. One has d^{ω} ∘ d^{ω} = 0 and if d^{ω}π = 0, π is said to be d^{ω}-closed. If ω is exact, then π is said to be d^{ω}-exact.

Let ω ∈ ⋀^{1}(M). If ω ≠ 0, ω = ∧dω = 0, then ω is called a class 1 1-form.

If M is endowed with a 2-form Ω, then one can define the morphism

Ω^{♭} : T_{x}M → T_{x}^{∗}M

as Ω^{♭}(Z) = ^{♭}Z = −i_{Z}Ω; Z ∈ T_{x} = M, for all x ∈ M, where

i_{Z}Ω(X) = Ω(Z, X); ∀X ∈ T_{x}M.

We shall denote by grad (resp. div) the gradient (resp. divergence) operator. Let G ∈ ⋀^{2m+1} M be the Riemannian volume element of M. We denote by ∗

the Hodge star operator of (M, g) with respect to G. We recall that the codifferential operator δ (see e.g. [?] p.153) is expressed by

δπ = (−1)^{n(r+1)+1} ∗ d ∗ π, π ∈ ⋀ ^{r}M.

Definition 1.1 [?] A function f : R^{n} → R is isoparametric if ∥gradf∥^{2} and div (gradf) can be expressed as functions of f.

Now, we shall remember some basic results about almost contact manifolds (see e. g. [?], [?]).

Definition 1.2 A differentiable manifold M is said to have an almost contact structure if admits a vector field ξ (called the Reeb vector field), a 1-form η and a (1, 1) tensor field ϕ satisfiying

\begin{matrix}η(ξ) = 1; \\ ϕ^{2} = −I + η ⊗ ξ \end{matrix} (1.2)

The above conditions imply that M is odd dimensional, say 2m + 1, and ϕ(ξ) = 0; rankϕ = 2m; η ∘ ϕ = 0. The distribution kerη is called horizontal and Span{ξ} is called vertical.

Definition 1.3 A manifold endowed with an almost contact structure (ϕ, ξ, η) and with a Riemannian metric g satisfying

g(ϕX, ϕY) = g(X, Y) − η(X)η(Y), (1.3)

for any vector fields X, Y ∈ X M is called almost conctact Riemannian manifold.

If M is an almost contact Riemannian manifold, then the 2-form Ω defined by

Ω(X, Y) = g(ϕX, Y); X, Y ∈ X M; (1.4)

satisfies η ∧ Ω^{m} ≠ 0, which implies that M is orientable. Moreover, rank Ω = 2m and i_{ξ}Ω = 0.

Note that, on an almost contact Riemannian manifold, one has η(X) = g(X, ξ); ∀X ∈ X M; i.e; ξ = η^{♯}. In the above conditions, it is easily seen that

g(Z, ϕZ′) + g(ϕZ, Z′) = 0; ∀Z, Z′ =∈ X M. (1.5)

Remark 1.4 Let (M, ϕ, η, ξ, g) be an almost contact Riemannian manifold and x ∈ M. Then one can choose a local field of orthonormal frames over a neighborhood of x

O = {e_{μ}, μ = 0, 1, …, 2m} = {ξ, e_{1}, …, e_{m}, ϕe_{1}, …, ϕe_{m}}

and its associated coframe O^{∗} = {ω^{0}, ω^{1}, …, ω^{m}, ω^{m+1}, …, ω^{2m}}, where ω^{0} = η. Then, with respect to this frame

ϕ = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & −I_{m} \\ 0 & I_{m} & 0 \end{matrix} )

i.e., ϕ = −ω^{a^{∗}} ⊗ e_{a} + ω^{a} ⊗ e_{a^{∗}} where a ∈ {1, …, m} and a^{∗} = a + m.

We shall use the capital (resp. greek) letters when the subindex runs in {1, … 2m} (resp {0, 1, … 2m}).

Then one has

Ω = ∑_{a=1}^{m} ω^{a} ∧ ω^{a^{∗}}; I = η ⊗ ξ + ω^{A} ⊗ e_{A}.

Observe that: X = ∑_{μ=0}^{2m} X^{μ}e_{μ} ⇒ X^{♭} = ∑_{μ=0}^{2m} X^{μ}ω^{μ}.

With respect to O and O^{∗} E. Cartan’s structure equations can be written as

∇e_{λ} = θ_{λ}^{μ} ⊗ e_{μ} (1.6)

dω^{λ} = −θ_{μ}^{λ} ∧ ω^{μ} (1.7) dθ_{μ}^{λ} = −θ_{μ}^{γ} ∧ θ_{γ}^{λ} + Θ_{μ}^{λ} (1.8)

In the above equations θ (resp. Θ) are the local connection forms in the bundle O(M) (resp. the curvature 2-forms on M).

From the orthonormality of the frame O and ∇g = 0 one can deduce that the matrix

θ = ( \begin{matrix}θ_{0}^{0} & … & θ_{2m+1}^{0} \\ ⋮ & ⋮ & ⋮ \\ θ_{0}^{2m+1} & … & θ_{2m+1}^{2m+1} \end{matrix} )

is hemisymmetric, i.e., θ_{μ}^{λ} = −θ_{λ}^{μ}. In particular θ_{μ}^{μ} = 0.

2 Closed concircular almost contact Riemannian manifolds

We introduce the following notion:

Definition 2.1 Let (M, ϕ, η, ξ, g) be an almost contact Riemannian manifold. A vector field U is called the generative of ξ if U verifies the following two conditions

i) ξ is U-concircular, i.e; ∇ξ = η ⊗ U, (2.9)

ii) U^{♭} is closed, i.e; dU^{♭} = 0, (2.10)

Then the manifold (M, ϕ, η, ξ, g, U) is called a closed concircular almost contact Riemannian manifold (abr. CCACR manifold).

Now we shall prove some equations and identities, which will be used in this work.

Lemma 2.2 Let (M, ϕ, η, ξ, g, U) be a CCACR manifold. Then:

(1) U = ∇_{ξ}ξ ∈ kerη and ∇_{U}ξ = 0.

(2) θ_{0}^{A} = U^{A}η; θ_{0}^{0} = 0.

(3) the next Kähler relations are satisfied:

θ_{b}^{a} = θ_{b^{∗}}^{a^{∗}}; θ_{b}^{a^{∗}} = θ_{a}^{b^{∗}}; a ∈ {1, … m}; a^{∗} = a + m.

(4) The covariant differential of any vector field Z on M is expressed by

∇Z = (dZ^{A} + Z^{B}θ_{B}^{A}) ⊗ e_{A} + (dZ^{0} − g(Z, U) = η) ⊗ ξ + Z^{0}η ⊗ U.

(5) For any vector field Z on M, we have

(∇ϕ)Z = −η(Z)η ⊗ ϕU + g(ϕU, Z)η ⊗ ξ;

where ((∇ϕ)Z)(X) = ϕ∇_{X}Z − ∇_{X}ϕZ; ∀X =∈ X M.

Proof. (1) By (??) and (??)

(∇ξ)(ξ) = (η ⊗ U)(ξ) ⇒ ∇_{ξ}ξ = η(ξ)U = U.

Let X be a vector field on M. We have 0 = (∇_{X}g)(ξ, ξ) ⇒ g(∇_{X}, ξ) = 0. Then ∇_{X}ξ ∈ Span{ξ}^{⊥} = kerη; ∀X ∈ X M. In particular U = ∇_{ξ}ξ ∈ kerη. Then by (??) one obtains ∇_{U}ξ = η(U)U = 0.

(2) Set U = U^{A}e_{A}. Let X be a vector field on M. By 1) ∇_{X}ξ ∈ kerη. One has ∇_{X}ξ = θ_{0}^{0}(X)ξ + θ_{0}^{A}(X)e_{A}. Then θ_{0}^{0}(X) = 0; ∀X ∈ X M.

If X ∈ kerη, by (??) ∇_{X}ξ = 0. Then θ_{0}^{A}(X) = 0; ∀X ∈ kerη.

If X = ξ, U = θ_{0}^{A}(ξ)e_{A} = U^{A}e_{A}.

From the above equations one can states θ_{0}^{A} = U^{A}η.

(3) See e. g. [?], [?].

(4) Let Z, X ∈ X M and set Z = Z^{0}ξ + Z^{A}e_{A}. Then, one has ∇_{X}Z = ∇_{X}Z^{0}ξ + ∇Z^{A}e_{A}.

Using (??) one derives that ∇_{X}Z^{0}ξ = (dZ^{0} ⊗ ξ)(X) + (Z^{0}η ⊗ U)(X).

From (??) and the identities θ_{A}^{0} = −θ_{0}^{A} and θ_{0}^{A} = U^{A}η one can states

∇_{X}Z^{A}e_{A} = ((dZ^{A} + Z^{B}θ_{B}^{A}) ⊗ e_{A})(X) − (g(Z, U)η) ⊗ ξ)(X).

Then, one can obtains

∇Z = (dZ^{A} + Z^{B}θ_{B}^{A}) ⊗ e_{A} + (dZ^{0} − g(Z, U) = (η) ⊗ ξ + Z^{0}η ⊗ U.

(5) Using 3) and 4) and by a straightforward calculation, one can state

(∇ϕ)Z = −η(Z)η ⊗ ϕU + g(ϕU, Z)η ⊗ ξ,

thus finishing the proof. □

We shall give a sufficient condition which implies that U^{♭} is an exact form.

Remark 2.3 Let Y be any kerη vector field on M, and assume that Y defines a skew-symmetric Killing vector field (abr. SSK) on M having the structure vector field U as generative (see [?]). Then one may write

∇Y = U^{♭} ⊗ Y − Y^{♭} ⊗ U. (2.11)

Since U is also a kerη vector field, it follows by Lemma ?? (4) that necesarily g(U, Y) = 0, and with the help of the structure equations (??) and (??) one derives by a standard calculation dY^{♭} = 2U^{♭} ∧ Y^{♭}. (2.12)

This proves that Y^{♭} is an exterior recurrent form which has 2U^{♭} as recurrence form (see [?]). We notice that (??) is the standard equation of SSK vector fields (see [?]).

In addition setting 2l = ∥Y ∥^{2} and taking account that g(U, Y) = 0 one gets from (??) dl = 2lU^{♭} (2.13)

which shows that the existence of Y implies that U^{♭} be an exact form.

Finally, we shall prove that the existence of such a vector field Y is assured by an exterior differential system in involution. If we denote by Σ the exterior differential system which defines Y, it follows (see [?]) by E. Cartan’s that the characteristic numbers are r = 3, s_{0} = 1, s_{1} = 2. Since r = s_{0} + s_{1} it follows that Σ is an involution and the existence of Y depends of 2 arbitrary functions of 1 argument.

Now we shall obtain the main properties of η, Ω and U in a CCACR manifold.

Proposition 2.4 (Properties of η.) Let (M, ϕ, η, ξ, g, U) be a CCACR manifold. Then

(1) dη = U^{♭} ∧ η, and then η is a exterior recurrent.

(2) δη = 0.

(3) d^{−U^{♭}}η = 0.

(4) dη ≠ 0, η ∧ dη = 0, i.e; η is a class 1 1-form.

Proof. (1) Taking account θ_{0}^{A} = U^{A}η (see Lemma ?? (2)) and applying the equation (??) to ω^{0} = η one finds

dη = −θ_{B}^{0} ∧ ω^{B} = θ_{0}^{B} ∧ ω^{B} = U^{B}η ∧ ω^{B} = −U^{B}ω^{B} ∧ η = −U^{♭} ∧ η;

(which says that η is exterior recurrent 1-form and has the closed 1-form U^{♭} as recurrence form [?]).

(2) By definition of δ one has δη = − ∗ d ∗ η and on behalf of Lemma ?? (2) one derives δη = 0, which afirms that η is a coclosed form.

(3) On the other hand by the cohomological operator d^{ω} one may also write the equation dη = U^{♭} ∧ η as d^{−U^{♭}}η = 0, and say that η is d^{−U^{♭}}-closed.

(4) In consequence of 1) and 2) η is a coclosed exterior recurrent form. In addition by Lemma ?? (2) one may write dη ≠ 0; η ∧ dη = 0, these equations proves that η is of class 1 1-form. □

Remark 2.5 We consider the flow F defined by the unitary vector field ξ. By Prop. ??, (1) we have that the distribution kerη = Span{ξ}^{⊥} is involutive. Remember that dη ≠ 0, then by virtue of [?] Theorem 10.6 we concludes that U_{x} = (∇_{ξ}ξ)_{x} ≠ 0; ∀x ∈ M, and that the vector field ξ is not geodesible, i.e; the orbits of ξ are not geodesics on M.

Corollary 2.6 Let (M, ϕ, η, ξ, g, U) be a CCACR manifold, then M is foliated by 1-codimensional almost Hermitian submanifolds.

Proof. Let x ∈ M, we consider N the maximal integral submanifold verifiying that x ∈ N; T_{y}N = kerη_{y} ∀y, ∈ N.

We defined J ∈ T_{1}^{1}(M) by J = ϕ|_{N}. Then, by (??) J^{2}(X) = −X; ∀X ∈ X N, and then (N, J, g|_{N}) is an almost Hermitian manifold. □

Proposition 2.7 Let (M, ϕ, η, ξ, g, U) be a CCACR manifold and let Ω the 2-form defined by (??), then dΩ = 0.

Proof. Due to M has an almost contact Riemannian structure the horizontal connection forms θ_{B}^{A} satisfy the Kähler relations of Lemma ?? (3). By Remark ?? one has Ω = ∑ w^{a} ∧ w^{a^{∗}}, then by a standard calculation one finds using θ_{0}^{A} = U^{A}η and Prop. ?? (4), dΩ = 0. □

Remark 2.8 Since U^{♭} = ∑_{A=1}^{2m} U^{A}ω^{A} is a closed form, taking into account (??) one

derives

∑_{A,B=1}^{2m}(dU^{A} + U^{B}θ_{B}^{A}) ∧ ω^{A} = 0.

If there exists λ ∈ F(M) such that

dU^{A} + U^{B}θ_{B}^{A} = λω^{A}; ∀A ∈ {1, … 2m};

the above equation is satisfied.

This remark allows us to introduce the following

Definition 2.9 Let (M, ϕ, η, ξ, g, U) be a CCACR manifold. We say that M is a λ-CCACR manifold if there exists λ ∈ F(M) such that

dU^{A} + U^{B}θ_{B}^{A} = λω^{A}; ∀A ∈ {1, … 2m}. (2.14)

In the rest of the paper, we shall work with λ-CCACR manifolds, although some results will be also true for general CCACR manifolds.

Proposition 2.10 ( Properties of U.) Let (M, ϕ, η, ξ, g, U) be a λ-CCACR manifold. Then

(1) L_{U}η = ∥U∥^{2}η and L_{U}Ω = 2λΩ; λ = const., and then U is an infinetesimal conformal transformation of the almost contact structure.

(2) (∇ϕ)U = 0.

(3) ∇_{U}U = λU; i.e., U is an affine geodesic.

(4) [U, ξ] = ∥U∥^{2}ξ, which shows that ξ admits an infinitesimal transformation of generators U.

(5) g(∇_{Z}U, Z′) = g(∇_{Z′}U, Z); ∀Z, Z′ ∈ X M. i.e., U is a kerη-gradient vector field.

Proof. (1) Recall that ^{♭}U = ∑(U^{a^{∗}}ω^{a} − U^{a}ω^{a^{∗}}). Next making use of (??) one gets

d(^{♭}U) = −2λΩ (2.15)

and consequently since Ω is closed one has λ = const. and so one infers L_{U}Ω = 2λΩ. Moreover since η(U) = 0, one derives from Prop. ??, (1) L_{U}η = ∥U∥^{2}η and by the two last obtained equations we conclude that the generative U of ξ defines and infinitesimal conformal transformation of the almost contact structure of M.

(2) (∇ϕ)U = 0 can be obtained quickly from Lemma ?? (5), taking Z = U.

(3) Using Lemma ?? (4) and taking Z = U vector field U is expressed by

∇U = (dU^{A} + U^{B}θ_{B}^{A}) ⊗ e_{A} + (−g(U, U)η) ⊗ ξ.

Then by (??)

∇U = λω^{A} ⊗ e_{A} − ∥U∥^{2}η ⊗ ξ = λdp − (λ + ∥U∥^{2})η ⊗ ξ. (2.16)

Since η(U) = 0, the above equation implies ∇_{U}U = λU. Moreover ∇_{ξ}U = −∥U∥^{2}ξ. (4) Because ∇ is symmetric T (ξ, U) = 0. Then [ξ, U] = −∥U∥^{2}ξ.

(5) It also should be noticed that by (??) one has : g(∇_{Z}U, Z′) = g(∇_{Z′}U, Z); for any vector fields Z, Z′ on M, which afirms that U is a gradient vector field (in the sense of Okumura; also [?] and by d^{−U^{♭}}η = 0 one may say that η is a d^{−U^{♭}}-exact 1-form). □

Proposition 2.11 Let (M, ϕ, η, ξ, g, U) be a λ-CCACR manifold.

(1) If D_{U} = Span{U, ξ} then M may be viewed as the locally product of integral submanifolds of D_{U} and D_{U}^{⊥}.

(2) Let Y be any kerη SSK vector field on M. If D_{Y} = Span{U, Y, ξ} then D_{Y} defines a foliation on M. Moreover, U defines an infinitesimal conformal transformation of Y and [ξ, Y] = 0.

Proof. (1) By Prop. ?? (4), D_{U} is an involutive distribution. Let U′, U′′ be vector fields of D_{U}. By Lemma ?? and Prop. ?? one has:

∇_{ξ}ξ = U; ∇_{ξ}U = −∥U∥^{2}ξ; ∇_{U}ξ = 0; ∇_{U}U = λU;

and then it is easily seen that ∇_{U′}U′′ ⊂ D_{U}, and consequently the leaves of D_{U} are totally geodesic.

The vector fields ξ and U are not singular and then the distributions Span{ξ} = kerη and Span{ξ}^{⊥} = kerU^{♭} are well defined. Both distributions are involutive because dη = U^{♭} ∧ η (see Prop. ?? (1)), and dU^{♭} = 0 (see equation (??)), and then D_{U}^{⊥} = kerη ∩ kerU^{♭} is also involutive.

(2) We consider the 3-form φ = U^{♭} ∧ Y^{♭} ∧ η; it is easily seen by (??) and Prop. ??

(1) that dφ = 0, and consequently the 3-distribution D_{Y} = Span{U, Y, ξ} defines a foliation. Since i_{Y} φ = ∥Y ∥^{2}U = ∧φ one also derives L_{Y} φ = 0, that is φ is invariant by Y.

By (??), (??) and (??) a short calculation gives [U, Y] = (∥U∥^{2}−λ)Y and [ξ, Y] = 0 which says that U defines an infinitesimal conformal transformation of Y, and ξ and Y commute. □

Proposition 2.12 Let (M, ϕ, η, ξ, g, U) be a λ-CCACR manifold. The transversal curvature forms Θ_{0}^{A}(A ∈ {1, …, 2m}) are monomial an expressed by

Θ_{0}^{A} = (U^{A}U^{♭} + λω^{A}) ∧ η

and the vector valued 2-form Π = ∑ Θ_{0}^{A} ⊗ e_{A} = −λη ∧ dp + (U^{♭} ∧ η) ⊗ U is d^{∇}-closed, and U defines an almost conformal transformation of Π, i.e.,

L_{U}Π = ∥U∥^{2}Π + 2λ(U^{♭} ∧ η) ⊗ U.

Proof. With the help of the structure equations (??) corresponding to the curvature forms Θ, one derives by (??) and (??)

Θ_{0}^{A} = (U^{A}U^{♭} + λω^{A}) ∧ η (2.17)

and this shows that Θ_{a}^{A} are monomial 2-forms.

Further by (??) we deduce the following vector valued 2-form Π = ∑ Θ_{0}^{A} ⊗ e_{A} = −λη ∧ dp + (U^{♭} ∧ η) ⊗ U and we agree to denominate Π the structure curvature vector valued form. Now operating on Π by the exterior covariant derivate operator d^{∇}, and since d(U^{♭} ∧ η) = 0 (see Prop. ??, (1)), a short calculation gives d^{∇}Π = 0. Hence Π enjoys the significative property to be a closed vector valued form. Taking now the Lie derivative of Π with respect to U , one finds by Lemma ?? (3) L_{U}Π = −λL_{U}η ∧ dp + L_{U}(U^{♭} ∧ η) ⊗ U = ∥U∥^{2}Π + 2λ(U^{♭} ∧ η) ⊗ U.

3 Harmonic Properties

In this section we shall discuss some harmonic properties induced by the almost contact structure of the λ-CCACR manifold M under consideration.

Note that ∗U^{♭} = ∑(−1)^{A}ω^{1} ∧ … ∧ ω̂^{A} ∧ … ∧ ω^{2m} ∧ η and then one finds that the codifferential δU^{♭} of U^{♭} is expressed by δU^{♭} = ∥U∥^{2} − 2mλ.

Hence since dU^{♭} = 0, the harmonic operator ΔU^{♭} is expressed by

ΔU^{♭} = λ∥U∥^{2}U^{♭}.

This is known afirms that U^{♭} is an eigenfunction of Δ, having λ∥U∥^{2} as eigenvalue.

By similar devices and since it has been shown in section 2 that η is coclosed one finds by Prop. ??, (1) that

Δη = 2mλη.

Hence η is also an eigenfunction of Δ. Next according to the concept of conformal adjoint transformation, one has regarding the paring (η, U) L_{U}^{∗}η = δ(U^{♭} ∧ η) + U^{♭} = ∧δη. Then since η is coclosed, one derives by Prop. ??, (1)

L_{U}^{∗}η = − 2mλη,

that is U defines a infinitesimal homothety of the conformal adjoint L^{∗}η.

Let now L be the Weil’s (1, 1) operator [?] defined by L : π → π ∧ Ω, π ∈ ⋀^{2} M. (note that one has dLπ = Ldπ + π ∧ L = π), and set η_{q} = η ∧ Ω^{q}.

Then by Prop. ?? (1) one gets dη_{q} = U^{♭} ∧ η_{q} and

L_{ξ}η_{q} = −U^{♭} ∧ Ω^{q} = −(U^{♭})_{q}

i.e., the Lie derivatives of the (2q +1)-form η_{q} are equated (up to the sign) by (2q +1)- forms (U^{♭})_{q}.

Finally in an other order of ideas, one finds by Prop. ??, (4)

∥grad (∥U∥^{2})∥^{2} = 4λ^{2}∥U∥^{2}.

Now by a short calculation one derives div U = 2mλ − ∥U∥^{2} and

div grad (∥U∥^{2}) = 8mλ^{2} − 4λ∥U∥^{2}.

Since λ = const., we may afirm that ∥U∥^{2} is an isoparametric function.

Then we can state:

Theorem 3.1 Let M(ϕ, η, ξ, U, g) be a λ-CCACR manifold . Then the tuple of structure tensors (ξ, U, η) satisfies the following properties:

(1) The 1-forms U^{♭} and η are eigenfunctions of Δ and U defines an infinitesimal homothety of the conformal adjoint 2m-form ^{∗}η.

(2) The Lie derivatives with respect to ξ of the (2q + 1)-forms η_{q} = η ∧ Ω^{q} are expressed by

L_{ξ}η_{q} = −U^{♭} ∧ Ω^{q}.

(3) ∥U∥^{2} is an isoparametric function.

4 ϕ-soldering vector fields

Taking into account Lemma ?? (4) one has

∀X ∈ kerη; ∇X = (dX^{A} + X^{B}θ_{B}^{A}) ⊗ e_{A} − g(X, U)η ⊗ ξ.

Then we introduce the following

Definition 4.1 Let (M, ϕ, η, ξ, g, U) be a λ-CCACR manifold. A ϕ-soldering vector field is a nonsingular vector field X ∈ kerη satisfying

∇X = fϕdp − g(X, U)η ⊗ ξ, f ∈ F(M) (4.18)

We study the main properties of such a vector field:

Theorem 4.2 Let (M, ϕ, η, ξ, g, U) a (2m + 1)-dimensional λ-CCACR manifold endowed with a ϕ-soldering vector field X. Then:

(1) X defines an infinitesimal automorphism of Ω which moves to a potential form.

(2) f is a constant function and ϕX defines an infinitesimal homotethy of Ω.

(3) X is a Kerη Killing vector field, i.e.,

g(∇_{Z}X, Z′) + g(∇_{Z′}X, Z) = 0

for any η(Z) = 0, η(Z′) = 0.

(4) X and ϕX commute.

(5) ϕX is a kerη gradient vector field. i.e.,

g(∇_{Z}ϕX, Z′) = g(∇_{Z′}ϕX, Z)

for any Z, Z′ ∈ X M.

(6) X and ϕX define infinitesimal conformal transformations of ξ.

(7) The sectional curvature of the plane spanned by X and ϕX vanishes.

(8) M is locally the product of a 3-dimensional submanifold M_{X} tangent to D_{X} = Span{ϕX, ξ, U} and of a (2m − 2)-dimensional submanifold M_{X}^{⊥} tangent to D_{X}^{⊥}.

Proof. (1) By Lemma ?? (4) and (??) we find

\{ \begin{matrix}dX^{a} + X^{A}θ_{A}^{a} = −fω^{a^{∗}} \\ dX^{a^{∗}} + X^{A}θ_{A}^{a^{∗}} = fω^{a}, a ∈ \{1, …, m\}, a^{∗} = a + m \end{matrix} \} (4.19)

and setting 2ℓ = ∥X∥^{2}, β = (ϕX)^{♭} one derives from (??) dℓ = −fβ and dβ = 0. Note that

β(Y) = g(ϕX, Y) = Ω(X, Y); −^{♭}X(Y) = (i_{X}Ω)Y = Ω(X, Y);

for any vector field Y on M. Then, one has β = −^{♭}X.

Recall that dΩ = 0 (see Prop. ??), then one may write

L_{X}Ω = (i_{X} ∘ d + d ∘ i_{X})Ω = d(−^{♭}X) = dβ = 0.

(2) One has X^{♭} = ∑_{A=1}^{2m} X^{A}ω^{A}. With the help of (??) one deduces that dX^{♭} = −fΩ

wich implies by Prop. ?? that f = constant.

As X ∈ kerη; one obtains X^{♭} = −^{♭}(ϕX) = i_{ϕX}Ω, and then

L_{ϕX}Ω = (i_{ϕX} ∘ d + d ∘ i_{ϕX})Ω = dX^{♭} = −fΩ.

(3) By (??) one finds g(∇_{Z}X, Z′)+g(∇_{Z′}X, Z) = 2λη(Z)η(Z′) for any vector fields Z, Z′ on M. Therefore if Z, Z′ ∈ Kerη one may write g(∇_{Z}X, Z′) + g(∇_{Z′}X, Z) = 0. In this case we agree to say that X is a kerη Killing vector field.

(4) Recall that (∇ϕ)Z = −η(Z)η ⊗ ϕ(U) + g(ϕU, Z) for any Z vector field on M (see Lemma ?? (5)).

Making use of the equations (??) and (??) one has

∇_{Y} ϕX = ϕ(∇_{Y} X) − (∇ϕ)(X)(Y) = −fY + fη(Y)ξ − g(ϕU, X)η(Y)ξ,

for any vector field Y on M. Then one infers

∇ϕX = −fdp + (f − g(ϕU, X))η ⊗ ξ. (4.20)

Then by (??) and the above equation T (X, ϕX) = 0 ⇒ [X, ϕX] = 0, which shows that X and ϕX commute.

(5) From (??) one may write

g(∇_{Z}ϕX, Z′) = g(∇_{Z′}ϕX, Z), ∀Z, Z′ ∈ kerη

which says that ϕX is a kerη gradient vector field.

(6) Taking into account that ξ is U-concircular, X is a ϕ-soldering vector field and the equation (??) one finds [X, ξ] = g(X, U)ξ; [ϕX, ξ] = g(ϕU, X)ξ; and so following the above calculations prove that both X and ϕX define infinitesimal conformal transformations of ξ.

(7) Operating on (??) by the operator d^{∇} one gets by a standard calculation

d^{∇}(∇X) = ∇^{2}X = ((λU^{♭} − f(ϕU)^{♭} ∧ η) ⊗ ξ.

By (??) one has

K_{ϕX∧X} = \frac{g(R(ϕX, X)X, ϕX)}{∥X∥^{2}∥ϕX∥^{2} − g(X, ϕX)^{2}} = \frac{g(((λU^{♭} − f(ϕU)^{♭} ∧ η) ⊗ ξ(X, ϕX), ϕX}{∥X∥^{2}∥ϕX∥^{2} − g(X, ϕX)^{2}}

where K_{ϕX∧X} is the sectional curvature of the plane spanned by X and ϕX. Then, a short calculation gives K_{ϕX∧X} = 0.

(8) By Lemma ?? (3), it is seen by similar consideration as in Prop. ??(1), then M is locally the product of a 3-dimensional submanifold M_{X} tangent to D_{X} = Span{ξ, U, ϕX} and of a (2m − 2)-dimensional submanifold M_{X}^{⊥} tangent to D_{X}^{⊥}. □

Acknowledgements. The first author is partially supported by Spanish Grant DGICYT PB 95-0124.

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Fernando Etayo, Rafael Santamaria Departamento de Matemáticas, Estadística y Computación Facultad de Ciencias Universidad de Cantabria Avda. de los Castros, s/n, 39071 Santander, SPAIN e-mail: etayof@matesco.unican.es, santamar@matesco.unican.es

Radu Rosca 59 Av. Emile Zola, 75015 Paris, FRANCE