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)
|
________________
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
Aq(M,TM) = Hom(ΛqTM,TM)
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
dp = ωA ⊗ e
A (1.1)
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 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
θBA = g(T ,e
B ∧ eA) = TBωA − TAωB (2.1)
( ∧: the
wedge product of vector fields).
It follows from (2.1)
∇TeA = 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
∇eA = TAdp − ωA ⊗T (2.3)
and
dTA = fωA, f ∈ Λ0N. (2.4)
It has been proved in [DR] that the forms
ωA of the
cobasis {ωA} = O∗
satisfy
dωA = α ∧ ωA, (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
NJ(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 = ∑
haa re
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 = −∑
rTre
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(AHea,eb) = −g(∇eAH,eb) = ∥T⊥∥2δ
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
hr are,
as is known [Ch], expressed by
habr = hr(e
a,eb) = g(h(ea,eb),er) = g(∇eaeb,er) = θbr(e
a) = −Trδ
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
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)
Θar = 0, Θ
sr = 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
Θba = −(2f + T2)ωa ∧ ωb. (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 = MD × MD⊥,
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