1. INTRODUCTION

Let (M,g) be a Riemannian (or pseudo-Riemannian) [C.sup.[infinity]]-manifold, and [nabla], d p, and [??] : TM [right arrow] [T.sup.*] 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

[nabla]X = [U.sup.[??]] [cross product] + [X.sup.[??]] [cross product] U, (1.1)

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.sup.[??] = 0).

If U is a closed torse forming [8,9]

[[nabla].sub.Z]U = aZ + g(Z, U)U, a = const., (1.2)

then the existence of such an X is determined by an exterior differential system in involution (in the sense of Cartan ) 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.sub.X] tangent to X and U.

If LU is the Lie derivative, we also find

[L.sub.U][[nabla].sub.U] = 0, [U,X] = aX, (1.3)

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

[nabla]V = X [conjunction] U,

([conjunction] : wedge product) and prove that V is a 2-exterior concurrent vector field. Finally two examples are given.

2. PRELIMINARIES

Let (M,g) be a Riemannian [C.sup.[infinity]]-manifold and [nabla] be the covariant differential operator with respect to the metric tensor g. We assume that M is oriented and [nabla] is the Levi-Civita connection. Let [GAMMA]TM be the set of sections of the tangent bundle and [??] : TM [right arrow] [T.sup.*]M and [??] = [[??].sup.-1] the classical musical isomorphisms defined by g.

As usual, we denote by [C.sup.[infinity]]M and [GAMMA][[LAMBDA].sup.1]TM the algebra of smooth functions on M and the set of 1-forms on M, respectively.

Following , we denote by [A.sup.q](M,TM) = [GAMMA]Hom([[LAMBDA].sup.q]TM, TM) the set of vector-valued q-forms, q < dimM, and by

[d.sup.[nabla]] : [A.sup.q](M,TM)[right arrow][A.sup.q+1](M,TM)

the covariant derivative operator with respect to [MATHEMATICAL EXPRESSION NOT ASCII.] The vector-valued 1-form d p [epsilon] [A.sup.1](M,TM) is the identity vector-valued 1-form, called the soldering form of M (see ). Since [nabla] is symmetric, we have [d.sup.[nabla]](d p) = 0. A vector field Y such that

[d.sup.[nabla]]([nabla]Y) = [[nabla].sup.2]Y = [pi] [conjunction]d p [member of] [A.sup.2] (M,TM) (2.1)

for some 1-form [pi] (called the concurrence form) is defined as exterior concurrent vector field [4,8]. If R is the Ricci tensor of [nabla], we have

R(Y,Z) = -(n-1)[lambda]g(Y,Z), Z [member of] [GAMMA]TM, (2.2)

where n = dimM and [pi] = [lambda][Y.sup.[??]] ([lambda] [member of] [C.sup.[infinity]] M is a conformal scalar).

A vector field U such that

[nabla]U = ad p + u [cross product] U, u [member of] [GAMMA][[LAMBDA].sup.1]TM, a ]member of] [C.sup.[infinity]]M, (2.3)

is called a torse forming .

Let O = {e.sub.A], A = 1, ..., n} be a local field of adapted vectorial frames over M and let [O.sup.*] = {[[omega].sup.A]} be its associated coframe. Then the soldering form d p of M is expressed by d p = [[omega].sup.A] [cross product] [e.sub.A] and Cartan structure equations written in an indexless manner are

[[nabla].sub.e] = [theta] [cross product] e, (2.4)

d [omega] = - [theta] [conjunction] [omega], (2.5)

d [theta] = -[theta] [conjunction] [theta] + [THETA]. (2.6)

In the above equations, [theta] (resp. [THETA]) are the local connection forms in the tangent bundle TM (resp. the curvature forms on M).

3. 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 [nabla]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 ).

In the present paper we consider a BC vector field X such that

[nabla]X = [U.sup.[??] [cross product] X + [X.sup.[??]] [cross product] (3.1)

where U is a certain vector field called the generative of X. It is easy to prove that

g([[nabla].sub.Z]X,Z') = g([[nabla].sub.Z],X.Z), Z,Z' [member of] [GAMMA]TM, (3.2)

which shows that X is a gradient vector field in the sense of Okumura (see ). Using Cartan's structure equations, it follows that

[dX.sup.[??]] = 0. (3.3)

In the current paper we assume that U is a closed torse forming , i.e.

[nabla]U = ad p + [U.sup.[??]] [cross product] U [right and left arrow], [[nabla].sub.Z] U = aZ + g(Z,U)U, a = const. (3.4)

From (3.1) and (3.4) we derive

[L.sub.U]X = ([[parallel]U[parallel].sup.2]-a)X, (3.5)

which, as is known, proves that X admits an infinitesimal transformation U.

Since

[dU.sup.[??]] = 0, (3.6)

it follows from (3.3) and (3.6) that M receives a foliation.

Operating on (3.1) and (3.4) by [d.sup.[nabla]], we derive by a standard calculation

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.7)

which proves that X and U are exterior concurrent vector fields. Then, by reference to , the Ricci tensors of X and U are expressed by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.8)

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 ).

It follows from (3.8) that if M is compact, then the constant a is positive. In order to simplify, we set

[l.sub.X] = [[parallel]X[parallel].sup.2], [l.sub.U] = [[parallel]U[parallel].sup.2], s = g(X,U). (3.9)

We obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.9)

Denote now by [SIGMA] 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 [SIGMA] (i.e. Cartan's numbers) are r = 5, [s.sub.0] = 3, [s.sub.1] = 2. Since r = [s.sub.0] + [s.sub.1], it follows that [SIGMA] 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.sub.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'' [member of] [D.sub.X] , then

[[nabla].sub.X'']X' [member of] [D.sub.X]. (3.11)

Summing up, we conclude from (3.11) and (3.8) that the manifold (M,g) carrying X is foliated by Einstein surfaces [M.sub.X] tangent to [D.sub.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 MX tangent to X and U.

In another order of ideas, if we take the Lie derivative of [nabla]U with respect to U and since a = const., we get

[L.sub.U][nabla]U = 0, (3.12)

which means that U is an affine vector field.

Further, we define a vector field V such that

[nabla]V = X [conjunction] U = [U.sup.[??]] [cross product] X - [X.sup.[??]] [cross product] U. (3.13)

We find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (3.15)

i.e., V is a 2-exterior concurrent vector field.

We also remark that V is a Killing vector field, i.e.

g([[nabla].sub.Z] V,Z') + g([[nabla].sub.Z']V,Z) = 0. (3.16)

From the general formula

[dV.sup.[??]](U,X) = g([[nabla].sub.U]V,X)-g(U,[[nabla].sub.X]V),

we also derive

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Next we consider the skew-symmetric Killing vector field W having U as generative , i.e.

[nabla]W =W [conjunction] U. (3.17)

Then, by Rosca's Lemma  it follows that

[dW.sup.[??]] = [aU.sup.[??]] [conjunction] (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 [conjunction] 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.

4. EXAMPLES

We shall determine the BC gradient vector fields on two Riemannian manifolds.

1. We take the upper half space [x.sup.n] > 0 in the sense of Poincare's representation as the model of the hyperbolic n-space form [H.sup.n]. The metric of [H.sup.n] is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The Christoffel's symbols with respect to g are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the other being zero.

The vector field [xi] = [x.sup.n] [partial derivative]/[partial derivative][x.sup.n] is a closed torse forming (see ).

We determine the BC gradient vector fields on [H.sup.n] having [xi] as generative. Equation (3.1) can be written as

[nabla]X = u [cross product] X + v [cross product] x,

where u = [[xi.sup.[??]] and v = [X.sup.[??]].

Let

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In particular, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or, equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

By integrating we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

On the other hand, for [mu] [member of] {1, ..., 1}, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or, equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

It follows that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

2. Let [T.sup.n-1] be an (n-1)-dimensional flat torus with the coordinate system ([x.sup.1],....,[x.sup.n-1]) and R a real line with coordinate [x.sup.n]. Consider the warped product [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]. Then the components of the Riemannian metric on M are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and Christoffel's symbols are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] the other being zero.

We can prove that [xi] = [partial derivative]/[partial derivative][x.sup.n] is a closed torse forming (see ). The BC gradient vector fields on M having [xi] as generative are defined by

[nabla]X = u [cross product] X + v [cross product] [xi],

with u = [[xi].sup.[??]] and v = [X.sup.[??]].

If we put

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

then v = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] .

We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or, equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For [mu] [member of] {1, ..., n-1}, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

i.e.,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

or, equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The last equation implies [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

ACKNOWLEDGEMENT

The author would like to remind that Prof. Radu Rosca (1908-2005) was a well-known mathematician working on structured Riemannian manifolds admitting distinguished vector fields. This paper was started at the advice of Prof. Rosca, to whom the author will always be indebted.

doi: 10.3176/proc.2009.3.03

Received 15 October 2008, revised 19 January 2009, accepted 21 January 2009

REFERENCES

[1.] Cartan, E. Systemes Differentiels Exterieurs et Leurs Applications Geometriques. Hermann, Paris, 1975.

[2.] Dieudonne, J. Elements d'Analyse, Vol. IV. Gauthier Villars, Paris, 1977.

[3.] Matsumoto, K., Mihai, A., and Rosca, R. Riemannian manifolds carrying a pair of skew symmetric Killing vector field. An. St. Univ. "Al. I. Cuza" Iasi, 2003, 49, 137-146.

[4.] Mihai, I., Rosca, R., and Verstraelen, L. Some Aspects of the Differential Geometry of Vector Fields. K.U. Leuven, K.U. Brussel, PADGE 2, 1996.

[5.] O'Neill, B. Semi-Riemannian Geometry. Academic Press, 1983.

[6.] Poor, W. A. Differential Geometric Structures. McGraw Hill, New York, 1981.

[7.] Reyes, E. and Rosca, R. On biconcircular gradient vector fields. Rend. Sem. Mat. Messina Serie II, 1999, 6 (21), 13-25.

[8.] Rosca, R. An exterior concurrent skew-symmetric Killing vector field. Rend. Sem. Mat. Messina, 1993, 2, 131-145.

[9.] Yano, K. On torse-forming directions in Riemannian spaces. Proc. Imp. Acad. Tokyo, 1984, 20, 340-345.

Title Annotation: Printer friendly Cite/link Email Feedback MATHEMATICS Mihai, Adela Proceedings of the Estonian Academy of Sciences Report 4EXES Sep 1, 2009 2285 Hypersurfaces with pointwise 1-type Gauss map in Lorentz--Minkowski space/ Huperpinnad punktiti 1-tuupi Gaussi kujutusega Lorentzi-Minkowski ruumis. Convergence theorems on generalized equilibrium problems and fixed point problems with applications/Uldistatud tasakaalu ja pusipunkti ulesannete... Manifolds (Mathematics) Theorems (Mathematics)