Lobachevskii Journal of Mathematics Vol. 13, 2003, ??? – ???

©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)

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

2000 Mathematical Subject Classification. 53C05, 53C15.

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

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 = MD × MD⊥, where MD is an invariant submanifold of M and MD⊥ 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 ΓTM and ♭ : TM → T∗M be the set of sections of the tangent bundle TM and the musical isomorphism [P] defined by g, respectively. Following [P], we set

A q (M, T M) = H o m (Λ q T M, T M)

and notice that elements of Aq(M,TM) are vector valued q-forms.

If p ∈ M, then the vector valued 1-form dp ∈ A1(M,TM) is the canonical vector valued 1-form of M. Since ∇ is symmetric, one has d∇(dp) = 0.

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

d p = ω A \otimes e A (1.1)

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

\nabla e = θ \otimes e, (1.2)

d ω = - θ \land ω, (1.3)

d θ = - θ \land θ + Θ . (1.4)

In the above equations, θ, resp. Θ, are the local connection forms in TM 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 J2 = −1 and a 2-form Ω of rank 2m and let T (TA), A = 1,...2m, be a globally defined vector field of components TA.

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

θ B A = g (T, e B \land e A) = T B ω A - T A ω B (2.1)

( ∧: the wedge product of vector fields).

It follows from (2.1)

\nabla 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

\nabla e A = T A d p - ω A \otimes T (2.3)

and

d T A = f ω A, f \in Λ 0 N . (2.4)

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

d ω A = α \land 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, J2 = −1, is defined to be a CR-submanifold of N if there exists on M a differentiable distribution D : p → Dp ⊂ TpM satisfying the following conditions:

i) D is invariant (or holomorphic), i.e. JDp = Dp;

ii) the complementary orthogonal distribution D⊥ : p → D p⊥ ⊂ T pM is antiinvariant, i.e. JDp⊥ ⊂ T p⊥M.

We define the following distributions:

Dp = TpM ∩ J(TpM),

Dp⊥ = {Z ∈ T pM∣g(Z,X) = 0,∀X ∈ Dp}.

By a standard calculation, it follows that

N J (Z, Z') = 0,

for any Z ∈ ΓD, Z′ ∈ ΓD⊥, where NJ 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 = MD × MD⊥ ( MD is a leaf of D and MD⊥ 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 habr the components of the second fundamental form, the mean curvature vector field H associated with the immersion x : M → N is expressed by

H = \sum h a a r e r m' . (2.7)

We recall that H is an extrinsic invariant.

By (2.1), one has

haar = g(h(e a,ea),er) = g(∇eaea,er) = g(θaC(e a)eC,er) =

= θar(e a) = Taωr(e a) − Trωa(e a) = −m′Tr.

Then

H = - \sum 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

∇eaH = −∇ea(Tre r) = −ea(Tr)e r − Tr∇ eaer = (2.9)

= −ea(Tr)e r − Trθ rb(e a)eb = −dTr(e a)er −∑ r(Tr)2e a = −∥T⊥∥2e a.

It follows at once from (2.9)

g (A H e a, e b) = - g (\nabla e A H, e b) = ∥ T ⊥ ∥ 2 δ a b . (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 hr are, as is known [Ch], expressed by

h a b r = h r (e a, e b) = g (h (e a, e b), e r) = g (\nabla e a e b, e r) = θ b r (e a) = - T r δ a b, (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 2-forms Θa r, Θs 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 Θb a of the CR-submanifold are given by

Θ b a = - (2 f + T 2) ω a \land 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 + T2 = const. (see [DR]).

Summing up, we state the following

Theorem. Let x : M → N be an immersion of a submanifold M in a 2m-dimensional 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 \times M D ⊥,

where MD is an invariant submanifold of M and MD⊥ 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