IJMMS 29:10 (2002) 573–577
PII. S0161171202007822
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
A CHARACTERIZATION OF HARMONIC FOLIATIONS
BY THE VOLUME PRESERVING PROPERTY
OF THE NORMAL GEODESIC FLOW
HOBUM KIM
Received 22 July 2001
We prove that a Riemannian foliation with the flat normal connection on a Riemannian
manifold is harmonic if and only if the geodesic flow on the normal bundle preserves the
Riemannian volume form of the canonical metric defined by the adapted connection.
2000 Mathematics Subject Classification: 53C12, 53D25.
1. Introduction. Let (M, gM ) be a Riemannian manifold. A foliation Ᏺ on M is Riemannian and gM bundle-like if all the leaves are locally equi-distant to each other.
Such a foliation is characterized by the property that a geodesic orthogonal to the
foliation at one point is orthogonal everywhere. For a Riemannian foliation, considerable efforts have been made to give global characterizations of the property that it is
harmonic, that is, all of its leaves are minimal submanifolds. For examples, a Riemannian foliation is harmonic if and only if either one of the following conditions holds:
(1) it is an extremal of the energy functional for special variations (see [2]); (2) it is
an extremal of the energy of the foliation under certain variations of the Riemannian
metric of the manifold (see [1]). In this paper, we give a dynamical characterization
of the harmonicity of a Riemannian foliation which has the flat normal connection in
the sense of Oshikiri [4].
Let Ᏺ be a Riemannian foliation of dimension p and codimension q on a Riemannian
manifold M of dimension n (p + q = n) with bundle-like metric gM . Throughout, we
work in the smooth category and the following notations are used:
• T M is the tangent bundle of M.
• L and L⊥ are the tangent bundle and the normal bundle of Ᏺ, respectively.
• Γ T M, Γ L, and Γ L⊥ are the spaces of sections of T M, L, and L⊥ , respectively.
• π : T M → L⊥ , π ⊥ : T M → L, and PᏲ : L⊥ → M are the canonical projections.
• ∇M is the Levi-Civita connection associated with gM .
Since Ᏺ is Riemannian, there exists a unique torsion-free metric connection ∇ on
L⊥ which is called adapted and given as follows (see [2]): for Z ∈ Γ L⊥ ,

π [X, Z] for X ∈ Γ L,
∇X Z =
π ∇M Z for X ∈ Γ L⊥ .
X
(1.1)
Associated with the above connection there is a bundle map CᏲ : T L⊥ → L⊥ called the
574
HOBUM KIM
connection map associated with Ᏺ given as follows. For ξ ∈ TZ L⊥ with (dPᏲ )(ξ) ≠ 0,
CᏲ (ξ) = ∇σ̇ (0) Z,
(1.2)
where Z is a curve in L⊥ such that d/dt|t=0 Z = ξ and σ (t) = PᏲ (Z(t)). This map gives
a metric g̃ on L⊥ defined by
g̃(ξ, η) = gM
dPᏲ Z (ξ), dPᏲ Z (η) + gM CᏲ (ξ), CᏲ (η)
(1.3)
for ξ, η ∈ TZ L⊥ . We denote the Riemannian volume form on L⊥ associated with g̃ by µ̃.
We define a local flow φt on L⊥ , called the normal geodesic flow of Ᏺ as follows. For
z ∈ L⊥ , let γ be a geodesic with initial velocity z. Since Ᏺ is Riemannian, γ̇(t) ∈ L⊥ for
each t in the domain of γ. We put φt (z) = γ̇(t) for z ∈ L⊥ and t in the domain of γ.
A foliation Ᏺ is said to have the flat normal connection if the normal bundle L⊥ of Ᏺ
admits an orthonormal frame field {Ep+1 , . . . , En } such that gM (∇M
Z Eα , Eβ ) = 0 for all
α, β = p + 1, . . . , n and all Z ∈ Γ L⊥ .
The purpose of this paper is to prove the following theorem.
Theorem 1.1. Let Ᏺ be a Riemannian foliation on a Riemannian manifold which
has a flat normal connection and µ̃ the Riemannian volume form on L⊥ corresponding
to g̃. Then Ᏺ is harmonic if and only if (φt ) preserves µ̃.
2. The proof. Let ζ be a vector field on L⊥ generated by the geodesic flow. It suffices
to show that Ᏺ is harmonic if and only if (Θζ µ̃)(z) = 0 at any given point z ∈ L⊥ , where
Θζ denotes the Lie derivative. Let {e1 , . . . , en } be an orthonormal basis of the tangent
space of M at the point m = PᏲ (z) such that ei ∈ Lm for i = 1, . . . , p and eα ∈ L⊥
m for
α = p +1, . . . , n. In a neighborhood of m, we may choose a frame {Eα : α = p +1, . . . , n}
of L⊥ , called an adapted frame, satisfying the following properties: Eα (m) = eα , α =
p +1, . . . , n, ∇eα Eβ = π (∇M
eα Eβ ) = 0 and ∇X Eα = π ([X, Eα ]) = 0 for any smooth section
X of L on U (see [3]). Since Ᏺ has the flat normal connection, we may choose Eα so
that ∇Eα Eβ = 0 for α, β = p +1, . . . , n. Completing this frame by an orthonormal frame
{Ei : i = 1, . . . , p} of L with Ei (m) = ei , we get a local orthonormal frame {E1 , . . . , En } of
H
for A = 1, . . . , n
T M on a neighborhood U of m with EA (m) = eA for A = 1, . . . , n. Let EA
⊥
be the horizontal lift of EA to T L , that is, the unique vector field on a neighborhood
H
H
) = EA and CᏲ (EA
) = 0, and EαV for α = p + 1, . . . , n the
of z in L⊥ such that dPᏲ (EA
vertical lift of Eα on T L⊥ , that is, the vector field on a neighborhood of z such that
H
H
V
dP (EαV ) = 0 and CᏲ (EαV ) = Eα . We put EA
(z) = eA
and EαV (z) = eα
. Now we compute
H V
V
, ep+1 , . . . , en
Θζ µ̃ (z) e1H , . . . , en
=−
p
H
H V
V
µ̃ e1H , . . . , ζ, EiH (z), . . . , epH , ep+1
, . . . , en
, ep+1 , . . . , en
i=1
−
n
H
H V
V
µ̃ e1H , . . . , epH , ep+1
, . . . , ζ, EαH (z), . . . , en
, ep+1 , . . . , en
α=p+1
−
n
α=p+1
H V
V
µ̃ e1H , . . . , en
, ep+1 , . . . , ζ, EαV (z), . . . , en
.
(2.1)
A CHARACTERIZATION OF HARMONIC FOLIATIONS . . .
575
But,
H
H V
V
, . . . , en
, ep+1 , . . . , en
µ̃ e1H , . . . , ζ, EiH (z), . . . , epH , ep+1
= g̃ ζ, EiH (z), eiH = gM dPᏲ ζ, EiH (m), ei ,
H
H V
V
µ̃ e1H , . . . , epH , ep+1
, . . . , ζ, EαH (z), . . . , en
, ep+1 , . . . , en
= gM dPᏲ ζ, EαH (z) , eα ,
(2.2)
where m = PᏲ (z) and α is the second fundamental form of Ᏺ (see [2]).
Let Wi be any vector field on M satisfying Wi (ϕti m) = ϕ̃ti z for the local flows (ϕti )
of Ei and (ϕ̃ti ) of EiH . From dPᏲ ◦ EiH = Ei ◦ PᏲ , we have PᏲ ◦ ϕ̃ti = ϕti ◦ PᏲ for any t.
Therefore,
i
d
i
ζ ϕ̃t (z)
|t=0 dPᏲ ◦ dϕ̃−t
dPᏲ ζ, EiH (z) =
dt
d
i
|t=0 dϕ−t
◦ dPᏲ ζ ϕ̃ti (z)
=
dt
d
i
|t=0 dϕ−t
=
◦ ϕ̃ti (z)
dt
i
d
i
|t=0 dϕ−t
Wi ϕt (m)
=
dt
= Wi , Ei (m).
(2.3)
Hence we have
gM dPᏲ ζ, EiH (z) , Ei (z) = gM Wi , Ei , Ei (m)
= gM Wi , ∇M
Ei Ei (m)
= gM Wi (m), α Ei , Ei (m)
= gM z, α Ei (m), Ei (m) .
(2.4)
Thus, we have
−
p
H
H V
V
, . . . , en
, ep+1 , . . . , en
µ̃ e1H , . . . , ζ, EiH (z), . . . , epH , ep+1
i=1
= −gM
p
α Ei (m), Ei (m)
z,
(2.5)
i=1
= −gM z, τ(m) ,
where τ(m) is the mean curvature vector of Ᏺ at m (see [2]).
On the other hand, we have
gM
dPᏲ ζ, EαH (m), eα = gM Wα , Eα (m), eα ,
(2.6)
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HOBUM KIM
where Wα is any vector field on M satisfying Wα (ϕtα m) = ϕ̃tα z for the local flows ϕtα of
Eα and ϕ̃tα of EαH , α = p + 1, . . . , n. Since Wα (ϕtα m) is an integral curve of EαH , we have
M
H
π (∇M
Eα Wα ) = CᏲ (Eα ) = 0. Moreover, by the choice of {Eα }, we have π (∇Wα Eα )(m) = 0.
Therefore,
gM
M
dPᏲ ζ, EαH (m), eα = gM ∇M
Wα Eα (m) − ∇Eα Wα (m), eα = 0.
(2.7)
Thus, to complete the proof, it suffices to show that
H V
V
µ̃ e1H , . . . , en
, ep+1 , . . . , ζ, EαV (z), . . . , en
= 0,
(2.8)
gM CᏲ ζ, EαV (z) , eα = 0.
(2.9)
that is,
For this purpose, we introduce a local coordinate system around a point z ∈ L⊥ as
follows: let (x A )A=1,...,n : U → Rn be a distinguished chart on a neighborhood U of m ∈
M. To z ∈ PᏲ−1 (U ) with PᏲ (z) = m, we assign (x 1 (m), . . . , x n (m), zp+1 (m), . . . , zn (m))
n
as its coordinates, where z = α=p+1 zα (m)Eα (m). Let γ be a geodesic orthogonal to
the leaves of Ᏺ and (x A (t) : A = 1, . . . , n) its local coordinates.
Write
γ̇(t) =
n
zα (t)Eα γ(t) .
(2.10)
α=p+1
By the choice of {Eα }, we get
d α
z =0
dt
(2.11)
for α = p + 1, . . . , n. Moreover, if we express Eα as Eα =
a smooth function on U , we have
n
A
A
A=1 fα (∂/∂x ),
n n
n
n
∂
d A
∂
= γ̇ =
z α Eα =
zα fαA
.
x
A
A
dt
∂x
∂x
A=1
α=p+1
α=p+1 A=1
where fαA is
(2.12)
Equations (2.10) and (2.11) imply that (x A (t), zα (t)) satisfy
n
d A
x =
zα fαA ,
dt
α=p+1
d α
z = 0.
dt
(2.13)
It follows that ζ can be locally expressed as
ζ=
α,A
zα fαA
∂
.
∂x A
A simple computation using the above expression of ζ gives
∂
ζ, EαV = −
fαA + zβ Eα fβA
.
A
∂x
A
β
(2.14)
(2.15)
577
A CHARACTERIZATION OF HARMONIC FOLIATIONS . . .
It is easy to show that for a vector field ξ =
is given by
CᏲ (ξ) =
α
ξ̃ +
α
where z =
αz
α
Eα and ∇∂A Eα =
A
(∂/∂x A ) +
α ξ̃
α
(∂/∂zα ), CᏲ (ξ)
Eα ,
(2.16)
β,A
γ
γ=p+1 ΓαA Eγ .
δ,σ ,A
Aξ
α β A
ΓβA
z ξ
n
CᏲ ζ, EαV = −
fαA +
Therefore,
A δ σ
z Eα fβ
Γσ A z Eδ .
β
(2.17)
β
But ΓσδA = 0 on U for A = 1, . . . , n and δ, σ = p+1, . . . , n by the choice of the frame {EA }.
Hence CᏲ ([ζ, EαV ]) = 0 and the proof is complete.
Acknowledgments. The present work was supported by the Basic Science Research Institute Program, Ministry of Education, 1995, BSRI-95-1422 and Yonsei University Faculty Grant for 1994.
References
[1]
[2]
[3]
[4]
M. D. Hvidsten and P. Tondeur, A characterization of harmonic foliations by variations of
the metric, Proc. Amer. Math. Soc. 98 (1986), no. 2, 359–362.
F. W. Kamber and P. Tondeur, Harmonic foliations, Harmonic Maps (New Orleans, La, 1980),
Lecture Notes in Math., vol. 949, Springer-Verlag, Berlin, 1982, pp. 87–121.
F. W. Kamber, P. Tondeur, and G. Tóth, Transversal Jacobi fields for harmonic foliations,
Michigan Math. J. 34 (1987), no. 2, 261–266.
G. Oshikiri, Some remarks on minimal foliations, Tôhoku Math. J. (2) 39 (1987), no. 2,
223–229.
Hobum Kim: Department of Mathematics, Yonsei University, Seoul 120-749, Korea
E-mail address: kimhb@yonsei.ac.kr