Iranian Journal of Mathematical Sciences and Informatics
Vol. 11, No. 1 (2016), pp 13-26
DOI: 10.7508/ijmsi.2016.01.002
Tangent Bundle of the Hypersurfaces in a Euclidean Space
Sharief Deshmukh
∗
, Suha B. Al-Shaikh
Department of Mathematics, College of Science, King Saud University,
P. O. Box # 2455, Riyadh-11451, Saudi Arabia.
E-mail: shariefd@ksu.edu.sa
E-mail: sbshaikh1@hotmail.com
Abstract. Let M be an orientable hypersurface in the Euclidean space
R
2n
with induced metric g and T M be its tangent bundle. It is known
that the tangent bundle T M has induced metric
g as submanifold of the
Euclidean space R
4n
which is not a natural metric in the sense that the
submersion π : (T M,
g) → (M, g) is not the Riemannian submersion. In
this paper, we use the fact that R
4n
is the tangent bundle of the Eu-
clidean space R
2n
to define a special complex structure J on the tangent
bundle R
4n
so that (R
4n
,
J,h, i) is a Kaehler manifold, where h, i is the
Euclidean metric which is also the Sasaki metric of the tangent bundle
R
4n
. We study the structure induced on the tangent bundle (T M,
g)
of the hypersurface M , which is a submanifold of the Kaehler manifold
(R
4n
, J,h, i). We show that the tangent bundle T M is a CR-submanif ol d
of the Kaehler manifold (R
4n
, J,h, i). We find cond ition s under which
certain special vector fields on the tangent bundle (T M,
g) are Killing
vector fields. It is also shown that the tangent bundle T S
2n−1
of the
unit sphere S
2n−1
admits a Riemannian metric g and that there exists a
nontrivial Killing vector field on the tangent bundle (T S
2n−1
, g).
Keywords: Tangent bundle, Hypersurface, Kaehler manifold, Almost contact structure, Kil lin g
vector field, CR-Submanifold, S e c on d fundamental form, Wiengarten map.
2000 Mathematics subject classification: 53C42, 53C56, 53D10.
∗
Correspond ing Author
Received 29 May 2013; Accepted 15 November 2015
c
2016 Academic Center for Education, Culture and Research TMU
13
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14 S. Deshmukh, S. B. Al-Shaikh
1. Introduction
Recently efforts are made to study the geometry of the tangent bundle of a
hypersurface M in the Euclidean space R
n+1
( cf. [3]), where the authors have
shown that the induced metri c on it s tangent bun dl e T M as submanifold of
the Eucli de an space R
2n+2
is not a natural metric. In [4], we have extended
the study initiated in [3] on the geometry of the tangent bundle T M of an
immersed orient abl e hypersurface M in the Euclidean space R
n+1
. It is well
known that Killing vector fields play an important role in shaping the geometry
of a Riemannian mani fol d , for instance the presence of nonzero Killing vector
field on a compact Riemannian manifold forces i t s Ricci curvature to be non-
negative and t hi s in particular implies that on a compact Riemannian manifolds
of negative Ricci curvature there does not exist a nonzero Killing vector field.
The study of Killing vector fields becomes more interesting on the tangent
bundle T M of a Riemannian manifold (M, g) as the tangent bundle T M is
noncompact. It is known that if the tangent bundle T M of a Riemannian
manifold (M, g) is equipped with Sasaki metric, then the verticle lift of a parallel
vector field on M is a Kil l i ng vector field (cf. [15]). However if the Sasaki metric
is replaced by the Cheeger-G r omol l metric, then the vertical lift of any nonzero
vector field on M is never Killing (cf. [14]). Note that both Sasaki metric as well
as Cheeger-Gromoll metrics ar e natural metrics. We consider an orientable real
hypersurface M of the Euclidean space R
2n
with th e induced metric g. Then
as the tangent bundle T M of M is a submanifold of codimension two in R
4n
,
it has induced metric
g and this metric g on T M is not a n atu r al metric as
the submersion π : (T M, g) → (M, g) is not the Riemannian submersion (cf.
[3]). Let N be the unit normal vector field to the hypersurface M and J be
the natural complex structure on t h e Euclidean space R
2n
. Then we have a
globally defined unit vector field ξ on the hypersurface given by ξ = −JN called
the characteristic vector field of the real hyper s ur fac e (cf. [1, 2, 5, 6, 7, 8, 9]),
and this vect or field ξ gives rise to two vector fields ξ
h
(the horizontal lift)
and ξ
v
(the ver ti c al lift) on the tangent bundle (T M,
g). In this paper, we
use the fact that R
4n
is the tan gent bundle of the Euclidean space R
2n
and
that the projection π : R
4n
→ R
2n
is a Riemannian submersion, to define
a special almost complex structure J on the tangent bundle R
4n
which is
different from the canonical complex structure of the Euclidean space R
4n
and
show that (R
4n
, J,h, i) is a Kaehler manifold, wher e h, i is the Euclidean metric
on R
4n
. It is shown that the codimension two submanifold (T M,
g) of the
Kaehler manifold (R
4n
, J,h, i) is a CR-submanifold (cf. [10]) and it naturally
inherits certai n special vector fields other than ξ
h
and ξ
v
, and in this paper we
are interested in fin di n g conditions under which these special vector fields are
Killing vector fields on (T M,
g). One of the interesting outcome of this study
is, we have shown that the tan gent bundle T S
2n−1
of the unit sphere S
2n−1
as
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Tangent Bundle of the Hypersurfac e s in a Euclidean Sp ac e 15
submanifold of R
4n
admits a nontrivial Killing vector field. It is worth pointing
out that on the tangent bundl e T S
2n−1
with Sasakian metric no vertical or
horizontal lift of a vector field is Killing as this will require the corresponding
vector field on S
2n−1
is parallel which is impossible as S
2n−1
is space of constant
curvature 1. Note that on even dimensional Riemannian manifolds which are
irreducible, it is difficul t to find Killing vector fields , where as on product s like
S
2k−1
× S
2l−1
, S
2k−1
× R
2l−1
, R
2k−1
× R
2l−1
one can easily find Killing vector
fields. Since the tangent bundle T S
2n−1
is trivial for n = 1, 2, 4, finding Killing
vector fi el ds is easy in these dimensions, but for n ≥ 5, it is not trivial.
2. Preliminaries
Let (M, g) be a Rieman ni an manifold and T M be it s tangent bundle with
projection map π : T M −→ M . Then for each (p, u) ∈ T M , the tangent space
T
(p,u)
T M = H
(p.u)
⊕V
(p,u)
, where V
(p,u)
is the kernel of dπ
(p,u)
:T
(p,u)
(T M) −→
T
p
M and H
(p.u)
is the kernel of the connection map K
(p,u)
: T
(p,u)
(T M) −→
T
p
M with respect to the Riemannian connection on (M, g). The subspaces
H
(p.u)
, V
(p,u)
are called the horizontal and vertical subspaces respectively. Con-
sequently, the Lie algebra of smooth vector fields X(T M) on the tangent bundle
T M admits the decomposition X(T M ) = H ⊕ V where H is called the hori zon -
tal distribution an d V is calle d the vertical distribution on th e tangent bundle
T M. For each X
p
∈ T
p
M, the horizontal lift of X
p
to a point z = (p, u) ∈ T M
is the unique vector X
h
z
∈ H
z
such that dπ(X
h
z
) = X
p
◦ π and the vertical
lift of X
p
to a point z = (p, u) ∈ T M is the unique vector X
v
z
∈ V
z
such that
X
v
z
(df) = X
p
(f) for all functions f ∈ C
∞
(M), where df is the func t ion defined
by (df)(p, u) = u(f ). Also for a vector field X ∈ X(M ), the horizontal lift of X
is a vector field X
h
∈ X(T M ) whose value at a point (p, u) is the horizontal lif t
of X(p) to (p, u) , the vertical lift X
v
of X is defined similarly. For X ∈ X(M )
the horizontal and vertical l i f t s X
h
, X
v
of X are uniquely det er mi n ed vector
fields on T M satisfying
dπ(X
h
z
) = X
π(z)
, K(X
h
z
) = 0, dπ(X
v
z
) = 0, K(X
v
z
) = X
π(z)
Also, we have for a smo ot h function f ∈ C
∞
(M) and vector fields X, Y ∈
X(M), that (f X)
h
= (f ◦ π)X
h
, (f X)
v
= (f ◦ π)X
v
, (X + Y )
h
= X
h
+ Y
h
and (X + Y )
v
= X
v
+ Y
v
. If dim M = m and (U, ϕ) is a chart on M with
local coordinates x
1
, x
2
, . . . , x
m
, then (π
−1
(U), ϕ) is a chart on T M wi t h lo-
cal coordinates x
1
, x
2
, . . . , x
m
, y
1
, y
2
, . . . , y
m
, where x
i
= x
i
◦ π and y
i
= dx
i
,
i = 1, 2, . . . , m.
A Riemannian metric
g on the tangent bun dl e T M is said to be natural met-
ric with respect to g on M if
g
(p,u)
(X
h
, Y
h
) = g
p
(X, Y ) and g
(p,u)
(X
h
, Y
v
) = 0,
for all vectors fields X, Y ∈ X(M) and (p, u) ∈ T M , that is th e projection map
π : T M −→ M is a Riemannian submersion.
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16 S. Deshmukh, S. B. Al-Shaikh
Let M be an orientable hypersurface of the Euclidean space R
2n
with im-
mersion f : M −→ R
2n
and T M be its tangent bundle . Then as F = df :
T M −→ R
4n
= T R
2n
is also an i mm er si on, T M is an immersed submanifold
of the Euclidean space R
4n
. We denote the induced metrics on M, T M by
g,
g respecti vely and the Euclidean metric on R
2n
as well as on R
4n
by h, i.
Also, we denote by
∇, ∇, D and D the Riemannian connections on M , T M,
R
2n
, and R
4n
respectively. Le t N and S be the unit normal vector field and
the shape operator of the hypersurface M . For the hypersurface M of the
Euclidean space R
2n
we have the following Gauss and Weingarten formulae
D
X
Y =
∇
X
Y + hS(X), Y i N , D
X
N = −S(X), X, Y ∈ X(M ) (2.1)
where S is the shape operator (Weingarten map). Similarly for the submanifold
T M of the Euclidean space R
4n
we have the Gauss and Weingarten formulae
D
E
F = ∇
E
F + h(E, F ), D
E
N = −
¯
S
N
(E) +
∇
⊥
E
N (2.2)
where E, F ∈ X(T M ) ,
∇
⊥
is the connection in the normal bundle of T M and
¯
S
N
denotes the Weingarten map in the di r ec ti on of the normal
N and is related
to the secon d fundame ntal form h by
h(X, Y ),
N
= g(
¯
S
N
(X), Y ) (2.3)
Also we observe that for X ∈ X(M) the vertical lift X
v
of X to T M ,
as X
v
∈ ker dπ, where π : T M → M is the natural submersion, we have
dπ(X
v
) = 0 that is df ( dπ(X
v
)) = 0 or equi val e ntly we get d(f ◦ π)(X
v
) = 0,
that is d(˜π ◦F )(X
v
) = 0 (
π : T R
2n
→ R
2n
), which gives dF (X
v
) ∈ ker d˜π =
¯
V.
Now we state the following results which are needed in our work.
Lemma 2.1. [3] Let N be the unit normal vector field to the hypers ur f ace M
of R
2n
and P = (p, X
p
) ∈ T M. Then t he horizontal and vertical lifts Y
h
P
, Y
c
P
of Y
p
∈ T
p
M satisfy
dF
P
(Y
h
P
) = (df
p
(Y
p
))
h
+ V
P
, dF
P
(Y
v
P
) = (df
p
(Y
p
))
v
where V
P
∈ V
P
is given by V
P
= hS
p
(X
p
), Y
p
i N
v
P
, N
v
P
being the vertical lift
of the unit normal N to with respect to the tangent bundle
π : R
4n
→ R
2n
.
Lemma 2.2. [3] If (M, g) is an orientable hypersurface of R
2n
, an d (T M, g)
is its tangent bundle as s u bman if ol d of R
4n
, then the metric
g on T M for
P = (p, u) ∈ T M , satisfies:
(i)
g
P
(X
h
P
, Y
h
P
) = g
p
(X
p
, Y
p
) + g
p
(S
p
(X
p
), u)g
p
(S
p
(Y
p
), u).
(ii)
g
P
(X
h
P
, Y
v
P
) = 0.
(ii)
g(X
v
, Y
v
) = g
p
(X
p
, Y
p
).
Remark 2.3. It is well known that a metr ic g defined on T M using the Rie-
mannian metric g of M (such as Sasaki metric, Cheeger-Gromoll metric) are
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Tangent Bundle of the Hypersurfac e s in a Euclidean Sp ac e 17
natural metrics in the sense that the submersion π : (T M,
g) −→ (M, g) b e-
comes a Riemannian submersion with respect to these metrics. However, as
seen from above Lemmas, the induced metric on the tangent bundle T M of a
hypersurface M of the Euclidean s pac e R
2n
, as a submanifold of R
4n
is not a
natural metric because of the present of the term g
p
(S
p
(X
p
), u)g
p
(S
p
(Y
p
), u) in
the inner product of horizontal vectors on T M . Moreover, note that the for an
orientable hypersurface M of the Euclidean space R
2n
, the vertical lift N
v
of
the u ni t norm al is tangential to the submanifold T M of R
4n
as seen in 2.1
In what follows, we drop the suffixes li ke in g
p
(S
p
(X
p
), u) and and it will be
understood from the context of the entities appearing in the equations.
Theorem 2.4. [3] Let (M, g) be an orientable hypersurface of R
2n
, and (T M, g)
be its tangent bundle as submanifold of R
4n
. If ∇ and
∇ denote the Riemann-
ian connections on (M, g) and (T M, g) respectively, then
(i)
∇
X
h
Y
h
= (
∇
X
Y )
h
−
1
2
(R(X, Y )u)
v
,
(ii)
∇
X
v
Y
h
= g(S(X), Y ) ◦ πN
v
(iii)
∇
X
v
Y
v
= 0, (iv)
∇
X
h
Y
v
= (
∇
X
Y )
v
+ g(S(X), Y ) ◦ π N
v
.
Lemma 2.5. [4] Let T M be the tangent bundle of an orientable hypersur f ace
M of R
2n
. Then for X, Y ∈ X(M ),
(i) h(X
v
, Y
v
) = 0,
(ii) h(X
v
, Y
h
) = 0,
(iii) h(X
h
, Y
h
) = g(S(X), Y ) ◦ π N
h
.
Lemma 2.6. [4] For the tangent bundle T M of an orientable hypersurface M
of R
2n
and X ∈ X(M ), we have
(i)
D
X
v
N
v
= 0,
(ii)
D
X
v
N
h
= 0,
(iii)
D
X
h
N
v
= −(S(X))
v
, (iv)
D
X
h
N
h
= −(S(X))
h
.
Let J be the natur al complex structure on the Euclidean space R
2n
, which
makes (R
2n
, J, h, i) a Kaehler manifold. Then on an orientable real hypersurface
M of R
2n
with unit normal N, we define a unit vector field ξ ∈ X(M ) by
ξ = −J N , with i t s dual 1-f or m η(X) = g(X, ξ), where g is the induced metric
on M. For X ∈ X(M ), we express JX = ϕ(X) + η(X)N, where ϕ(X) is t h e
tangential component of J X, and it follows that ϕ is a (1, 1) tensor field on M ,
and that (ϕ, ξ, η, g) defines an almost contact metric structure on M (cf. [5],
[8], [ 9] ) , that is
ϕ
2
X = −X + η( X)ξ, η(ξ) = 1, η ◦ ϕ = 0, ϕ(ξ) = 0
and
g(ϕX, ϕY ) = g(X, Y ) − η(X)η(Y ), X, Y ∈ X(M )
Moreover, we have the following.
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