Blair, David E.

Vanli, Aysel Turgut

Osaka City University Repository

Blair, D.E. and Turgut Vanli, A.
Osaka J. Math.
43 (2006), 193–200
CORRECTED ENERGY OF DISTRIBUTIONS FOR 3-SASAKIAN
AND NORMAL COMPLEX CONTACT MANIFOLDS
DAVID E. BLAIR and AYSEL TURGUT VANLI
(Received September 6, 2004)
Abstract
In this paper we show that the natural fibrations on 3-Sasakian manifolds and on
normal complex contact metric manifolds are minima of the corrected energy of the
corresponding distributions.
1. Introduction
In [6], Chaco´n, Naveira and Weston introduced the energy E(V) of a q-dimensional
distribution on a Riemannian manifold (M;g). They studied the first and second varia-
tion of the energy and as an application showed that the Hopf fibration S3 ,→ S4n+3 →
HP n is an unstable critical point. The corresponding result in the case of the energy
of a vector field for the Hopf fibration S1 ,→ S2n+1 → CP n is due to C.M. Wood
[14]. Wood showed that for n > 1, the critical point is unstable; for n = 1 Brito [4]
showed that this Hopf fibration is a minima.
Subsequently in [5], Chaco´n and Naveira introduced a corrected energy D(V) for
a q-dimensional distribution on a Riemannian manifold (M;g) and proved that D(V) is
≥ the integral of the sum of the mixed sectional curvatures associated to a compatible
basis. As a single application they showed that the Hopf fibration S3 ,→ S4n+3 →
HP n is a minimum of D(V). In the present paper we show that this application can
be greatly generalized to the natural fibrations on 3-Sasakian manifolds and on normal
complex contact metric manifolds.
2. Geometry of distributions
Let (Mn; g) be a compact oriented Riemannian manifold with a q-dimensional dis-
tribution or subbundle V and let H denote the orthogonal complementary distribution
of dimension p = n − q. Let {e1; : : : ; en} be a local orthonormal basis on Mn such
that {e1; : : : ; ep} span H and {ep+1; : : : ; en} span V and adopt the index conventions:
1 ≤ a; b ≤ n, 1 ≤ i; j ≤ p, p + 1 ≤ ;  ≤ n. The second fundamental form of the
horizontal distribution H in the direction e

and that of the vertical distribution V in
2000 Mathematics Subject Classification. Primary 53C25; Secondary 53C15, 53D10, 53E99.
194 D.E. BLAIR AND A. TURGUT VANLI
the direction e
i
are given respectively by
h

ij
= −g(∇
e
i
e

; e
j
)
; h
i

= −g(∇
e

e
i
; e

)
:
The mean curvature vectors of the horizontal and vertical distributions are given re-
spectively by
~
HH =
n∑
=p+1
(
1
p
p∑
i=1
h

ii
)
e

;
~
HV =
p∑
i=1
 1
q
n∑
=p+1
h
i


e
i
:
One can regard a distribution, such as V , as a section of the Grassmann bundle,
G(q;Mn), of oriented q-planes in the tangent spaces of Mn. The geometry of this
bundle was developed in [6]. We also view V as a map  : Mn → G(q;Mn) where
 (x) is a unit q-vector with respect to the induced metric on ∧q(Mn), in particular
 (x) = e
p+1(x) ∧ · · · ∧ en(x):
Note that we have chosen a local orthonormal basis; in [14] the variations of unit vec-
tor fields are through unit vector fields and the variations of distributions in [6] are
through unit q-vectors. The norm of the covariant derivative of  is given in terms of
the second fundamental forms of H and V by
(∗)
∑
a
∥∥∇
e
a

∥∥2 = ∑
i;j;
(
h

ij
)2
+
∑
i;;
(
h
i

)2
:
The energy of a distribution V was defined in [6] as
E(V) = 1
2
∫
M
n∑
a=1
∥∥∇
e
a

∥∥2
dvol +
n
2
vol(M):
The corrected energy of [5] is defined by
D(V) =
∫
M
(∑
a
∥∥∇
e
a

∥∥2 + p(p − 2)∥∥~HH∥∥2 + q2∥∥~HV∥∥2
)
dvol:
The main result of [5] is the following.
Theorem A. If V is integrable, then
D(V) ≥
∫
M
∑
i;

i
dvol
where 
i
is the sectional curvature of the plane section spanned by e
i
∈ H and
e

∈ V .
CORRECTED ENERGY OF DISTRIBUTIONS 195
3. 3-Sasakian manifolds
By a contact manifold we mean a differentiable manifold M2n+1 together with a
1-form  such that  ∧ (d)n 6= 0. It is well known that given  there exists a unique
vector field  , such that d(;X) = 0 and ( ) = 1;  is called the characteristic vector
field or Reeb vector field of the contact form .
A Riemannian metric g is an associated metric for a contact form  if, first of
all, (X) = g(X;  ) and secondly, there exists a field of endomorphisms  such that

2
= −I + ⊗  and d(X; Y ) = g(X;Y ). We refer to (; ; ; g) as a contact metric
structure and to M2n+1 with such a structure as a contact metric manifold.
An almost contact structure, (; ; ), consists of a field of endomorphisms , a
vector field  and a 1-form  such that 2 = −I +  ⊗  and ( ) = 1 and an almost
contact metric structure includes a Riemannian metric satisfying the compatibility con-
dition g(X; Y ) = g(X; Y ) − (X)(Y ).
The product M2n+1 × R carries a natural almost complex structure defined by
J
(
X; f
d
dt
)
=
(
X − f ; (X) d
dt
)
and the underlying almost contact structure is said to be normal if J is integrable. The
normality condition can be expressed as N = 0 where N is defined by
N(X; Y ) = [; ](X; Y ) + 2 d(X; Y );
[; ] being the Nijenhuis tensor of .
A Sasakian manifold is a normal contact metric manifold. In terms of the co-
variant derivative of  with respect to the Levi-Civita connection, the Sasakian condi-
tion is
(∇
X
)Y = g(X; Y ) − (Y )X:
As is well known, from this it is easily seen that
∇
X
 = −X
and in turn that  is a Killing vector field, i.e. the contact metric structure is K-contact.
It is also well known that on a K-contact manifold the sectional curvature of all plane
sections containing  are equal to +1 (see e.g. [1], p.92).
A manifold admitting three almost contact structures, (

; 

; 

),  = 1; 2; 3, sat-
isfying


= 



− 

⊗ 

= −



+ 

⊗ 

:


= 



= −



; 

= 

◦ 

= −

◦ 

196 D.E. BLAIR AND A. TURGUT VANLI
is said to have an almost contact 3-structure. Kuo [13] showed that given such a struc-
ture there exists a Riemannian metric g compatible with each of the three almost con-
tact structures giving us an almost contact metric 3-structure (

; 

; 

; g). If each
of the three structures is Sasakian we have a 3-Sasakian structure. A remarkable re-
sult of Kashiwada [11] is that if each of the three almost contact metric structures
(

; 

; 

; g) is a contact metric structure, then the structure is a 3-Sasakian structure.
There are many 3-Sasakian manifolds aside from the sphere S4n+3 including sever-
al homogeneous spaces; see e.g. [1] pp.218–220 or the survey of Boyer and
Galicki [3].
Using ∇
X


= −

X one readily obtains on a 3-Sasakian manifold that [

; 

] =
2

. Thus the distribution V determined by the tri-vector  = 

∧ 

∧ 

is integrable
with totally geodesic leaves. The horizontal distribution H is defined by 

= 0,  =
1; 2; 3.
We now state and prove the following theorem.
Theorem 1. The vertical distribution V on a compact 3-Sasakian manifold is a
minima of the corrected energy D(V).
Proof. The proof will be the direct computations of both sides of the inequal-
ity in Theorem A showing that they are equal. We first show that the mean curvature
vectors ~HV and ~HH vanish. This will follow immediately from the following compu-
tations:
h
i

= −g(∇


e
i
; 

) = g(e
i
;∇




) = g(e
i
;−



) = g(e
i
; 

) = 0;
h

ii
= −g(∇
e
i


; e
i
) = g(

e
i
; e
i
) = 0:
Thus by equation (∗) we have for the norm of the covariant derivative of the tri-vector
 defining V , ∑
a
∥∥∇
e
a

∥∥2 = ∑
i;j;
(
h

ij
)2
=
∑
i;j;
g
(∇
e
i


; e
j
)2
=
∑
i;
g(−

e
i
;−

e
i
) =
∑

4n = 12n:
On the other hand we have noted that on a Sasakian manifold all sectional curva-
tures of plane sections containg the characteristic vector field are equal to +1. Noting
this for each of the three Sasakian structures and the horizontal vectors e
i
, we have
that each 
i
= +1. Therefore ∑
i;

i
= 12n
and we see that both sides of the inequality are equal to 12n times the volume of the
manifold.
CORRECTED ENERGY OF DISTRIBUTIONS 197
4. Complex contact manifolds
A complex contact manifold is a complex manifold of odd complex dimension
2n + 1 together with an open covering {U} by coordinate neighborhoods such that
1. On each U , there is a holomorphic 1-form  with  ∧ (d)n 6= 0.
2. On U∩U ′ 6= ∅ there is a non-vanishing holomorphic function f such that  ′ = f  .
The complex contact structure determines a non-integrable distribution H by the
equation  = 0. A complex contact structure is given by a global 1-form if and only if
its first Chern class vanishes [2].
On the other hand let M be a Hermitian manifold with almost complex structure
J , Hermitian metric g and an open covering by coordinate neighborhoods {U}; M is
called a complex almost contact metric manifold if it satisfies the following two con-
ditions:
1. In each U there exist 1-forms u and v = u ◦ J , with dual vector fields U and
V = −JU and (1; 1) tensor fields G and H = GJ such that
H
2
= G
2
= −I + u⊗ U + v ⊗ V
GJ = −JG; GU = 0; g(X;GY ) = −g(GX;Y ):
2. On U ∩ U ′ 6= ∅, we have
u
′
= au− bv; v′ = bu + av
G
′
= aG− bH; H ′ = bG + aH
where a and b are functions on U ∩ U ′ with a2 + b2 = 1.
Since u and v are dual to the vector feilds U and V , we easily see from the sec-
ond condition that on U ∩U ′, U ′ = aU −bV and V ′ = bU +aV . Also since a2 +b2 = 1,
U
′∧V ′ = U∧V . Thus U and V determine a global vertical distribution V by  = U∧V
which is typically assumed to be integrable.
A complex contact manifold admits a complex almost contact metric structure for
which the local contact form  is u − iv to within a non-vanishing complex-valued
function multiple and the local tensor fields G and H are related to du and dv by
du(X; Y ) = g(X;GY ) + ( ∧ v)(X; Y ); dv(X; Y ) = g(X;HY )− ( ∧ u)(X; Y )
where  (X) = g(∇
X
U;V ), ∇ being the Levi-Civita connection of g (Ishihara and
Konishi [10], Foreman [7]). We refer to a complex contact metric manifold with a
complex almost contact metric structure satisfying these conditions as a complex con-
tact metric manifold.
Ishihara and Konishi [8], [9] introduced a notion of normality for complex contact
structures. Their notion is the vanishing of the two tensor fields S and T given by
S(X; Y ) = [G;G](X; Y ) + 2g(X;GY )U − 2g(X;HY )V + 2(v(Y )HX − v(X)HY )
+  (GY )HX −  (GX)HY +  (X)GHY −  (Y )GHX;
198 D.E. BLAIR AND A. TURGUT VANLI
T (X; Y ) = [H;H ](X; Y ) − 2g(X;GY )U + 2g(X;HY )V + 2(u(Y )GX − u(X)GY )
+  (HX)GY −  (HY )GX +  (X)GHY −  (Y )GHX:
However this notion is too strong; among its implications is that the underlying
Hermitian manifold (M;g) is Ka¨hler. Thus while indeed one of the canonical exam-
ples of a complex contact manifold, the odd-dimensional complex projective space, is
normal in this sense, the complex Heisenberg group, is not. In [12] B. Korkmaz gen-
eralized the notion of normality and we adopt her definition here. A complex contact
metric structure is said to be normal if
S(X; Y ) = T (X; Y ) = 0; for every X; Y ∈ H;
S(U;X) = T (V;X) = 0; for every X:
Even though the definition appears to depend on the special nature of U and V , it
respects the change in overlaps, U ∩ U ′, and is therefore a global notion. With this
notion of normality both odd-dimensional complex projective space and the complex
Heisenberg group with their standard complex contact metric structures are normal.
One important consequence of normality for us is that the sectional curvature of a
plane section spanned by a vector in V and a vector in H is equal to +1 (cf. Korkmaz
[12]). Another consequence of normality is that
(∗∗) ∇
X
U = −GX +  (X)V; ∇
X
V = −HX −  (X)U:
Theorem 2. If M is a compact normal complex contact metric manifold, then
the vertical distribution is a minima of the corrected energy, i.e.
D(V) =
∫
M
∑
i;

i
dvol
where 
i
is the sectional curvature of the plane section spanned by e
i
∈ H and
e

∈ V .
Proof. As with Theorem 1, the proof will be the direct computations of both
sides of the inequality in Theorem A showing that they are equal. We first note that
the integral submanifolds of V are totally geodesic. This follows readily from (∗∗)
and the fact G and H annihilate U and V . Therefore all hi

vanish and in particu-
lar ~HV = 0. Similarly
h
1
ii
= −g (∇
e
i
U; e
i
)
= −g (−Ge
i
+  (e
i
)V; e
i
) = 0
and
h
2
ii
= −g (∇
e
i
V; e
i
)
= −g (−He
i
−  (e
i
)U; e
i
) = 0
CORRECTED ENERGY OF DISTRIBUTIONS 199
which shows that the mean curvature vector ~HH vanishes.
Again by equation (∗) we have for the norm of the covariant derivative of the bi-
vector  = U ∧ V ,∑
a
∥∥∇
e
a

∥∥2 = ∑
i;j;
(
h

ij
)2
=
∑
i;j
[
g(∇
e
i
U; e
j
)2 + g(∇
e
i
V; e
j
)2]
=
∑
i;j
[
g(Ge
i
; e
j
)2 + g(He
i
; e
j
)2] = ∑
i
[g(Ge
i
;Ge
i
) + g(He
i
; He
i
)]
= 2
∑
i
g(e
i
; e
i
) = 8n:
On the other hand we have noted that on a normal complex contact metric man-
ifold all sectional curvatures of plane sections spanned by a vector in V and a vector
in H are equal to +1. Noting this for each of U;V ∈ V and the horizontal vectors e
i
,
we have that each 
i
= +1. Therefore∑
i;

i
= 8n
and we see that both sides of the inequality in Theorem A are equal to 8n times the
volume of the manifold.
References
[1] D.E. Blair: Riemannian Geometry of Contact and Symplectic Manifolds, Birkha¨user, Boston,
2002.
[2] W.M. Boothby: Homogeneous complex contact manifolds; in Proc. Sympos. Pure Math. III,
Amer. Math. Soc., 1961, 144–154.
[3] C.P. Boyer and K.Galicki: 3-Sasakian manifolds; in Essays on Einstein Manifolds, Surv. Differ.
Geom., VI, Int. Press, Boston, MA, 1999, 123–184.
[4] F.G.B. Brito: Total bending of flows with mean curvature correction, Differential Geom. Appl.
12 (2000), 157–163.
[5] P.M. Chaco´n and A.M. Naveira, Corrected energy of distributions on Riemannian manifolds,
Osaka J. Math. 41 (2004), 97–105.
[6] P.M. Chaco´n, A.M. Naveira and J.M. Weston: On the energy of distributions with application
to quaternionic Hopf fibrations, Monatsh. Math. 133 (2001), 281–294.
[7] B. Foreman: Variational Problems on Complex Contact Manifolds with Applications to Twistor
Space Theory, Thesis, Michigan State University, 1996.
[8] S. Ishihara and M. Konishi: Real contact 3-structure and complex contact structure, Southeast
Asian Bulletin of Math. 3 (1979), 151–161.
[9] S. Ishihara and M. Konishi: Complex almost contact manifolds, Ko¯dai Math. J. 3 (1980),
385–396.
[10] S. Ishihara and M. Konishi: Complex almost contact structures in a complex contact manifold,
Ko¯dai Math. J. 5 (1982), 30–37.
[11] T. Kashiwada: On a contact 3-structure, Math. Z. 238 (2001), 829–832.
200 D.E. BLAIR AND A. TURGUT VANLI
[12] B. Korkmaz: Normality of complex contact manifolds, Rocky Mountain J. Math. 30 (2000),
1343–1380.
[13] Y.-Y. Kuo: On almost contact 3-structure, Toˆhoku Math. J. 22 (1970), 325–332.
[14] C.M. Wood: On the energy of a unit vector field, Geom. Dedicata 64 (1997), 319–330.
David E. Blair
Department of Mathematics
Michigan State University
East Lansing, MI 48824, USA
e-mail: blair@math.msu.edu
Aysel Turgut Vanil
Department of Mathematics
Faculty of Science and Art
Gazi University
06500 Ankara, Turkey
e-mail: avanli@gazi.edu.tr


Corrected energy of distributions for 3-Sasakian and normal complex contact manifolds

Turgut Vanli, Aysel

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Corrected energy of distributions for 3-Sasakian and normal complex contact manifolds

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