Some properties of biconcircular gradient vector fields

We consider a Riemannian manifold carrying a biconcircular gradient vector field X , having as generative a closed torse forming U . The existence of such an X is determined by an exterior differential system in involution depending on two arbitrary functions of one argument. The Riemannian manifold is foliated by Einstein surfaces tangent to X and U . Properties of the biconcircular vector field X are investigated.


INTRODUCTION
Let (M, g) be a Riemannian (or pseudo-Riemannian) C ∞ -manifold, and ∇, d p, and : T M → T * M be the Levi-Civita connection, the soldering form of M (i.e. the canonical vector-valued 1-form of M), and the musical isomorphism defined by g, respectively.
A vector field X on M such that ∇X = U ⊗ X + X ⊗U, ( where U is a certain vector field, called the generative of X, is defined as a biconcircular gradient (abbr.BC gradient) vector field.In consequence of (1.1), X is a self-adjoint vector field (i.e., dX = 0).If U is a closed torse forming [8,9] ∇ Z U = aZ + g(Z,U)U, a = const., then the existence of such an X is determined by an exterior differential system in involution (in the sense of Cartan [1]) and depends on two arbitrary functions of one argument.In these conditions, we prove that a manifold (M, g) which carries such an X is foliated by Einstein surfaces M X tangent to X and U.
If L U is the Lie derivative, we also find i.e., U is an affine vector field and defines an infinitesimal homothety of X.
We also consider the skew-symmetric Killing vector field V defined by ∇V = X ∧U, (∧ : wedge product) and prove that V is a 2-exterior concurrent vector field.Finally two examples are given.

PRELIMINARIES
Let (M, g) be a Riemannian C ∞ -manifold and ∇ be the covariant differential operator with respect to the metric tensor g.We assume that M is oriented and ∇ is the Levi-Civita connection.Let ΓT M be the set of sections of the tangent bundle and : T M → T * M and = −1 the classical musical isomorphisms defined by g.
As usual, we denote by C ∞ M and ΓΛ 1 T M the algebra of smooth functions on M and the set of 1-forms on M, respectively.
Following [6], we denote by A q (M, T M) = ΓHom(Λ q T M, T M) the set of vector-valued q-forms, q < dim M, and by the covariant derivative operator with respect to ∇ (in general The vector-valued 1-form d p ∈ A 1 (M, T M) is the identity vector-valued 1-form, called the soldering form of M (see [2]).Since ∇ is symmetric, we have for some 1-form π (called the concurrence form) is defined as exterior concurrent vector field [4,8].
If R is the Ricci tensor of ∇, we have where n = dim M and π = λY (λ ∈ C ∞ M is a conformal scalar).
A vector field U such that is called a torse forming [9].
Let O = {e A ; A = 1, ..., n} be a local field of adapted vectorial frames over M and let O * = {ω A } be its associated coframe.Then the soldering form d p of M is expressed by d p = ω A ⊗ e A and Cartan structure equations written in an indexless manner are ) ) In the above equations, θ (resp.Θ) are the local connection forms in the tangent bundle T M (resp.the curvature forms on M).

PROPERTIES OF BICONCIRCULAR GRADIENT VECTOR FIELDS
A vector field X on a Riemannian (or pseudo-Riemannian) manifold (M, g) is said to be biconcircular (abbr.BC) if its covariant differential ∇X has no zero components only in two directions.
An example of a BC vector field is given by the skew-symmetric Killing vector field (in the sense of Rosca [8]).
In the present paper we consider a BC vector field X such that where U is a certain vector field called the generative of X.It is easy to prove that which shows that X is a gradient vector field in the sense of Okumura (see [7]).Using Cartan's structure equations, it follows that dX = 0. (3.3) In the current paper we assume that U is a closed torse forming [4], i.e.
From (3.1) and (3.4) we derive which, as is known, proves that X admits an infinitesimal transformation U.
which proves that X and U are exterior concurrent vector fields.Then, by reference to [8], the Ricci tensors of X and U are expressed by We recall that a (pseudo)-Riemannian manifold N is said to be Einstein if its Ricci tensor is given by R = cg, for some constant c (see [5]).
It follows from (3.8) that if M is compact, then the constant a is positive.In order to simplify, we set We obtain (3.10) Denote now by ∑ the exterior differential system which defines the BC gradient vector field X under consideration.
By (3.3), (3.6), and (3.10) the characteristic numbers of ∑ (i.e.Cartan's numbers) are r = 5, s 0 = 3, s 1 = 2. Since r = s 0 + s 1 , it follows that ∑ is in involution and by Cartan's test we conclude that the existence of X depends on two arbitrary functions of one argument.
Further, we denote by D X = {X,U} the 2-dimensional distribution spanned by X and U.
Since the property of exterior concurrency is invariant by linearity, it follows that if X , X ∈ D X , then Summing up, we conclude from (3.11) and (3.8) that the manifold (M, g) carrying X is foliated by Einstein surfaces M X tangent to D X .Theorem 1.Let (M, g) be a Riemannian manifold carrying a BC gradient vector field X with closed torse forming generative U.The existence of such an X is determined by an exterior differential system in involution depending on two arbitrary functions of one argument.
Any manifold (M, g) which carries such an X is foliated by Einstein surfaces M X tangent to X and U.
In another order of ideas, if we take the Lie derivative of ∇U with respect to U and since a = const., we get L U ∇U = 0, (3.12) which means that U is an affine vector field.Further, we define a vector field V such that i.e., V is a 2-exterior concurrent vector field.We also remark that V is a Killing vector field, i.e.
From the general formula we also derive Next we consider the skew-symmetric Killing vector field W having U as generative [3], i.e. ∇W = W ∧U. (3.17) Then, by Rosca's Lemma [8] it follows that dW = aU ∧W . (3.18) It should be noticed that, since a = const., [W,U] is also a Killing vector field.
Theorem 2. Let (M, g) be a Riemannian manifold carrying a BC gradient vector field X, having as generative a closed torse forming U. Then i) the generative U of the BC vector field X is an affine vector field; ii) the wedge product X ∧ U of X and U defines a 2-exterior concurrent vector field V, which is a Killing vector field; iii) if W is a skew-symmetric vector field having U as generative, then [W,U] is also a Killing vector field.

EXAMPLES
We shall determine the BC gradient vector fields on two Riemannian manifolds.
1. We take the upper half space x n > 0 in the sense of Poincaré's representation as the model of the hyperbolic n-space form H n .The metric of H n is given by The Christoffel's symbols with respect to g are the other being zero.
We can prove that ξ = ∂ ∂ x n is a closed torse forming (see [4]).The BC gradient vector fields on M having ξ as generative are defined by The last equation implies a = const.;then Bikaasringse gradientvektorvälja mõned omadused
) it follows from (3.3) and (3.6) that M receives a foliation.Operating on (3.1) and (3.4) by d ∇ , we derive by a standard calculation