Lobachevskii Journal of Mathematics Vol. 13, 2003, 81 – 85 \begin{matrix}c \\ ◯ \end{matrix}Koji Matsumoto, Adela Mihai, and Dorotea Naitza

Koji Matsumoto, Adela Mihai, and Dorotea Naitza

SUBMANIFOLDS OF AN EVEN-DIMENSIONAL MANIFOLD STRUCTURED BY A T -PARALLEL CONNECTION

(submitted by B. N. Shapukov)

Abstract. Even-dimensional manifolds N structured by a T -parallel connection have been defined and studied in [DR], [MRV].

In the present paper, we assume that N carries a (1,1)-tensor field J of square −1 and we consider an immersion x : M → N. It is proved that any such M is a CR-product [B] and one may decompose M as M = M_{D} × M_{D^{⊥}}, where M_{D} is an invariant submanifold of M and M_{D^{⊥}} is an antiinvariant submanifold of M.

Some other properties regarding the immersion x : M → N are discussed.

1. Preliminaries

Let (M, g) be a Riemannian C^{∞}-manifold and let ∇ be the covariant differential operator defined by the metric tensor g (∇ is the Levi-Civita connection).

Let ΓT M and ♭ : T M → T^{∗}M be the set of sections of the tangent bundle T M and the musical isomorphism [P] defined by g, respectively. Following [P], we set

A^{q}(M, T M) = Hom(Λ^{q}T M, T M)

and notice that elements of A^{q}(M, T M) are vector valued q-forms.

If p ∈ M, then the vector valued 1-form dp ∈ A^{1}(M, T M) is the canonical vector valued 1-form of M. Since ∇ is symmetric, one has d^{∇}(dp) = 0.

Let O = vect{e_{A}| A = 1, ..., 2m} be an adapted local field of orthonormal frames on M and O^{∗} = covect{ω^{A}} be its associate coframe. One has

dp = ω^{A} ⊗ e_{A}  (1.1)

CR-product.

Key words and phrases. T -parallel connection, space form, CR-submanifold,

2000 Mathematical Subject Classification. 53C05, 53C15.

This paper was written while the second author visited Yamagata University (Japan), supported by a JSPS postdoctoral fellowship.

82 KOJI MATSUMOTO, ADELA MIHAI, AND DOROTEA NAITZA

and Cartan’s structure equations, written in indexless manner, are:

∇e = θ ⊗ e,  (1.2)
dω = −θ ∧ ω,  (1.3)
dθ = −θ ∧ θ + Θ.  (1.4)

In the above equations, θ, resp. Θ, are the local connection forms in T M and the curvature forms on M, respectively.

2. CR-products

Manifolds structured by a T -parallel connection have been initiated by [R1] and several papers have treated such types of manifolds, as for instance [MRV], [DR].

Let N(J, Ω, g) be a 2m-dimensional C^{∞}-manifold endowed with a (1, 1)- tensor field J such that J^{2} = −1 and a 2-form Ω of rank 2m and let T (T^{A}), A = 1, ...2m, be a globally defined vector field of components T^{A}.

There, by reference [DR], one says that N is structured by a T -parallel connection if the connection forms θ_{B}^{A} and the vectors e_{A} of the orthonormal vector basis O = {e_{A}} satisfy

θ_{B}^{A} = g(T , e_{B} ∧ e_{A}) = T^{B}ω^{A} − T^{A}ω^{B}  (2.1)

(∧: the wedge product of vector fields).

It follows from (2.1)

∇_{T} e_{A} = 0,  (2.2)

which shows that all the vectors of O are T -parallel and this legitimates the definition of the structure of N.

In addition, it is shown in [DR] that one has

∇e_{A} = T^{A}dp − ω^{A} ⊗ T  (2.3)

and

dT^{A} = fω^{A}, f ∈ Λ^{0}N.  (2.4)

It has been proved in [DR] that the forms ω^{A} of the cobasis {ω^{A}} = O^{∗} satisfy

dω^{A} = α ∧ ω^{A},  (2.5)

where

α = T^{♭}.  (2.6)

In the present paper we will study submanifolds M of N.

Recall now the following definition [B]:

A submanifold M of a manifold N endowed with a (1, 1)-tensor field J, J^{2} = −1, is defined to be a CR-submanifold of N if there exists on M a differentiable distribution D : p → D_{p} ⊂ T_{p}M satisfying the following conditions:

i) D is invariant (or holomorphic), i.e. JD_{p} = D_{p};

ii) the complementary orthogonal distribution D^{⊥} : p → D_{p}^{⊥} ⊂ T_{p}M is antiinvariant, i.e. JD_{p}^{⊥} ⊂ T_{p}^{⊥}M.

We define the following distributions:

D_{p} = T_{p}M ∩ J(T_{p}M),
D_{p}^{⊥} = {Z ∈ T_{p}M| g(Z, X) = 0, ∀X ∈ D_{p}}.

By a standard calculation, it follows that

N_{J}(Z, Z′) = 0,

for any Z ∈ ΓD, Z′ ∈ ΓD^{⊥}, where N_{J} is the Nijenhuis tensor of J.

This result affirms that any submanifold M of a manifold N structured by a T -parallel connection is a CR-submanifold (see [B]).

By Frobenius theorem it is easily seen that both distributions D and D^{⊥} are integrable. Hence, following [B], such a CR-submanifold is defined as CR-product. In consequence of this fact, M is locally a Riemannian product M = M_{D} × M_{D^{⊥}} (M_{D} is a leaf of D and M_{D^{⊥}} is a leaf of D^{⊥}).

Consider now an m′-dimensional submanifold M of N and denote by q the dimension of the normal space corresponding to M, i.e. m′ + q = 2m. Then if a, b denote tangential indices and h_{ab}^{r} the components of the second fundamental form, the mean curvature vector field H associated with the immersion x : M → N is expressed by

H = \frac{∑ h_{aa}^{r} e_{r}}{m′} .  (2.7)

We recall that H is an extrinsic invariant.

By (2.1), one has

h_{aa}^{r} = g(h(e_{a}, e_{a}), e_{r}) = g(∇_{e_{a}}e_{a}, e_{r}) = g(θ_{a}^{C}(e_{a})e_{C}, e_{r}) =
= θ_{a}^{r}(e_{a}) = T^{a}ω^{r}(e_{a}) − T^{r}ω^{a}(e_{a}) = −m′T^{r}.

Then

H = − ∑_{r} T^{r}e_{r} = −T^{⊥}.  (2.8)

This says that, up to sign, H is expressed by the normal component of the structure vector field T .

Operating now on H by the covariant differential ∇ and taking account of (2.4), one infers

∇_{e_{a}}H = −∇_{e_{a}}(T^{r}e_{r}) = −e_{a}(T^{r})e_{r} − T^{r}∇_{e_{a}}e_{r} =  (2.9)

= −e_{a}(T^{r})e_{r} − T^{r}θ_{r}^{b}(e_{a})e_{b} = −dT^{r}(e_{a})e_{r} − ∑_{r}(T^{r})^{2}e_{a} = −∥T^{⊥}∥^{2}e_{a}.

It follows at once from (2.9)

g(A_{H}e_{a}, e_{b}) = −g(∇_{e_{A}}H, e_{b}) = ∥T^{⊥}∥^{2}δ_{ab}.  (2.10)

Hence, following a known definition [Ch], one may say that the immersion x : M → N is pseudo-umbilical. Moreover, since the second fundamental forms h^{r} are, as is known [Ch], expressed by

h_{ab}^{r} = h^{r}(e_{a}, e_{b}) = g(h(e_{a}, e_{b}), e_{r}) = g(∇_{e_{a}}e_{b}, e_{r}) = θ_{b}^{r}(e_{a}) = −T^{r}δ_{ab}, (2.11)

or equivalently

h(X, Y) = g(X, Y)H,  (2.12)

for any vector fields X and Y tangent to M.

This says that the immersion x : M → N is also totally umbilical.

Recall now that in general for any immersion x : M → N the curvature 2forms Θ_{a}^{r}, Θ_{s}^{r} are called the transversal and the vertical curvature forms [R2], respectively.

Taking use of (2.1) and (2.3), one finds with the help of the structure equations (1.3)

Θ_{a}^{r} = 0, Θ_{s}^{r} = 0.  (2.13)

This shows that the transversal and the vertical curvature forms associated with the immersion x : M → N vanish. In the same order of ideas one derives that the curvature forms Θ_{b}^{a} of the CR-submanifold are given by

Θ_{b}^{a} = −(2f + ∥T ∥^{2})ω^{a} ∧ ω^{b}.  (2.14)

Hence, following a well-known formula, the above relation affirms the relevant fact that the CR-submanifold M is a space form (see [KN], [YK]).

This also agrees the fact that 2f + ∥T ∥^{2} = const. (see [DR]).

Summing up, we state the following

Theorem. Let x : M → N be an immersion of a submanifold M in a 2mdimensional manifold N carrying a (1, 1)-tensor field J of square −1, structured by a T -parallel connection.

Then any such submanifold M is a CR-product and one may write

M = M_{D} × M_{D^{⊥}},

where M_{D} is an invariant submanifold of M and M_{D^{⊥}} is an antiinvariant submanifold of M.

In addition, one has the following properties:

(i) If T means the structure vector field on N, then the mean curvature vector field H associated with the immersion x : M → N is expressed by

H = −T^{⊥},

where T^{⊥} represents the normal component of T ;

(ii) The immersion x : M → N is pseudo-umbilical and totally umbilical; in particular, x′ : M_{D^{⊥}} → N is antiinvariant pseudo-umbilical and antiinvariant totally umbilical;

(iii) M is a space form submanifold of N.

Acknowledgements. The authors are very obliged to Prof. Dr. Radu Rosca for useful discussions and valuable advices.

References

[B] A. Bejancu: Geometry of CR-Submanifolds, D. Reidel Publ. Comp., Dordrecht, 1986

[Ch] B.Y. Chen: Geometry of Submanifolds, M. Dekker, New York, 1973

[DR] F. Defever, R. Rosca: On a class of even-dimensional manifolds structured by a T -parallel connection, Tsukuba J. Math. 25(2001), 359-369

[KN] S. Kobayashi, K. Nomizu: Foundations of Differential Geometry, vol. 2, Interscience, New York, 1969

[MOR] I. Mihai, A. Oiagă, R. Rosca, On a class of even-dimensional manifolds structured by an affine connection, Internat. J. Math. Math. Sci. 29 (2002), 681-686

[MRV] I. Mihai, R. Rosca, L. Verstraelen: On a class of exact locally conformal cosymplectic manifolds, Internat. J. Math. Math. Sci. 19 (1996), 247-278

[P] W. A. Poor: Differential Geometric Structures, Mc Graw Hill, New York, 1981

[R1] R. Rosca: On parallel conformal connections, Kodai Math. J. 2 (1979), 1-9

[R2] R. Rosca: On K-left invariant almost contact 3-structures, Results Math. 27 (1995), 117-128

[YK] K. Yano, M. Kon: Structures on Manifolds, World Scientific, Singapore, 1984.

Department of Mathematics, Faculty of Education, Yamagata University, 990-8560 Yamagata, Japan E-mail address: ej192@kdw.kj.yamagata-u.ac.jp Faculty of Mathematics, Str. Academiei 14, 70109 Bucharest, Romania E-mail address: adela@geometry.math.unibuc.ro Istituto di Matematica, Facoltà di Economia, Via dei Verdi 75, 98100 Messina, Italia

Received May 15, 2003