ON A CONSTRUCTION OF HARMONIC RIEMANNIAN SUBMERSIONS - csm.ro

ON A CONSTRUCTION
OF HARMONIC RIEMANNIAN SUBMERSIONS

MONICA ALICE APRODU

Using Killing vector fields, we present a (local) construction method of harmonic
Riemannian submersions similar to one previously developed by Bryant [8], [17].

AMS 2000 Subject Classification: 58E20, 53C43, 53C12.

Key words: Killing vector field, harmonic map, Riemannian submersion.

1. INTRODUCTION

Harmonic Riemannian submersions and, more generally, harmonic mor-
phisms as defined by Fuglede and Ishihara [15], [16], are solutions of an over-
determined differential system. For this reason, to give a general construction
method for harmonic morphism with fixed source manifold (M, gM ) is not an
easy task. Bryant [8] used Killing vector fields to contruct harmonic mor-
phisms with one-dimensional fibres, see also [17]. However, not all harmonic
morphisms of corank one are obtained in this way.

The aim of this short note is to give a construction method in the spirit
of [8] for maps of higher corank. For general facts about harmonic maps and
morphisms we refer to [3], [18], [10], [11], [12]; see [14] for an updated account
of Riemannian submersions.

In the last section we present some global examples obtained with the
help of the Boothby-Wang fibration of a regular Sasaki manifold, see [6], and
also [4], [5], [7].

2. THE GENERAL CONSTRUCTION

Let (M n+m, gM ) be a Riemannian manifold, and X1, . . . , Xm unitary
Killing vector fields on M , two-by-two orthogonal, such that
(1) [Xi, Xj] = 0,
for any i, j = 1, . . . , m. Suppose moreover (restricting to open subsets of M if
necessary) that X1, . . . , Xm are nowhere vanishing, hence linearly independent
at any point of M .

MATH. REPORTS 9(59), 3 (2007), 243–247

244 Monica Alice Aprodu 2

Equations (1) imply that the distribution generated by the given fields

is integrable, hence from the Frobenius Theorem (see for example [9]) it is
associated with an m-dimensional foliation F.

Let U ⊂ M be an open subset on which the foliation F is simple. In
other words, the leaves of F|U are the fibres of a submersion π : U → N n,
where N is an n-dimensional manifold.

For any i = 1, . . . , m, consider the one form ωi , gM -dual to the field Xi.
It is defined by ωi (∗) = gM (∗, Xi).

Lemma 1. With the notation as above, we have LXiωj = 0 for any
i, j = 1, . . . , m.

Proof. For any vector field Y on M , using (1) and the fact that Xi are
Killing, we obtain

(LXi ωj)(Y ) = Xiωj(Y ) − ωj([Xi, Y ]) = XigM (Y, Xj) − gM ([Xi, Y ], Xj) =
= gM (∇XiY, Xj ) + gM (Y, ∇Xi Xj ) − gM (∇XiY, Xj ) + gM (∇Y Xi, Xj ).

Hence

(LXi ωj)(Y ) = (LXi gM )(Xj , Y ) = 0.

Put m
(2)
gM = gM − (ωi )2.

i=1

From Lemma 1, for any i = 1, . . . , m we obtain LXigM = 0. Conse-
quently, there exists a Riemannian metric gN on N such that π∗(gN ) = gM .
By (2), we have the relation

m

gM = π∗(gN ) + (ωi )2,

i=1

hence the submersion π : (U, gM ) → (N, gN ) is horizontally conformal with
dilation function 1, i.e., it is a Riemannian submersion.

Proposition 2. The map π : (U, gM ) → (N, gN ) is harmonic.

Proof. Denoting by ωV the volume form on the fibres and using Propo-
sition 4.6.3, from [3], the harmonicity of π is equivalent to

LZ ωV = 0,

for any horizontal vector field Z. In our case, we have ωV = ω1 ∧ · · · ∧ ωm and

m

LZωV = ω1 ∧ · · · ∧ (LZ ωj) ∧ · · · ∧ ωm .

j=1

3 On a construction of harmonic Riemannian submersions 245

For any vertical vector fields Y1, . . . , Ym on M we have

m

(LZωV )(Y1, . . . , Ym) = (ω1 ∧ · · · ∧ (LZωj) ∧ · · · ∧ ωm)(Y1, . . . , Ym)

j=1

m

= ω1(Yσ(1)) . . . (LZ ωj)(Yσ(j)) . . . ωm(Yσ(m)).

j=1 σ∈Sm

Since Xi are Killing, and Yσ(j) can be written as

m

Yσ(j) = αij Xi

i=1

we can compute

(LZ ωj)(Yσ(j)) = dωj (Z, Yσ(j)) + Yσ(j)(ωj (Z))
= ZgM (Xj , Yσ(j)) − gM (Xj , [Z, Yσ(j)])

= gM (∇Z Xj , Yσ(j)) + gM (∇Yσ(j) Z, Xj )

m

= −2gM (∇Yσ(j) Xj , Z) = −2 αji gM (∇Xi Xj , Z).

i=1

If i = j, from [Xi, Xj] = 0 and the fact that Xi are Killing and ortho-
gonal, we obtain gM (∇XiXj, Z) = 0.

If i = j, the condition that Xi are Killing and unitary yields

gM (∇XiXi, Z) = 0.

Consequently, the map π is a harmonic Riemannian submersion, and the foli-
ation generated by the fields X1, . . . , Xm is a Riemannian foliation.

From Proposition 2 we obtain at once

Corollary 3. Let (M m+n, gM ) be a Riemannian manifold and X1, . . . ,
Xm unitary Killing vector fields, two-by-two orthogonal, and two-by-two com-
muting. Then the integral submanifolds of the distribution generated by X1, . . . ,
Xm are minimal.

3. GLOBAL EXAMPLES

The classical situation of the Hopf fibration can be used to obtain other
examples of Riemannian submersions by the construction of the previous sec-
tion. These examples are global ones whereas, in general, our construction will
work only locally.

It is well-known (cf. [5]) that the odd-dimensional sphere S2n+1 carries
a natural Sasaki structure. Let us denote by η the contact form and by ξ

246 Monica Alice Aprodu 4

the Killing vector field. Consider an integer s ≥ 1, let the Hopf fibration
p : S2n+1 → CPn, and define the iterated fibred product over CPn as

H2n+s = {(x1, . . . , xs) ∈ S2n+1 × · · · × S2n+1, p(x1) = · · · = p(xs)}.

Using the diagonal map

δ : CPn → CPn × · · · × CPn,

where the product is taken s times, and the natural projection

S2n+1 × · · · × S2n+1 → CPn × · · · × CPn,

one can interpret H2n+s as a principal torus bundle on CPn. Denote by
π : H2n+s → CPn the induced map. On H2n+s one defines for all i = 1, . . . , s
the 1-forms ηi = (0, . . . , η, . . . , 0) (η is on the ith position), and the dual vector
fields ξi = (0, . . . , ξ, . . . , 0). Note that η(ξ) = 1.

Using [4], one can verify that the vector fields ξi are Killing, unitary,
orthogonal, and satisfy (1). By construction, the distribution generated by
the vector fields ξi, i = 1, . . . , s, coincides with the vertical distribution of the
map π, hence the space of leaves is CPn. We deduce that π : H2n+s → CPn is
a harmonic Riemannian submersion.

Remark. The previous construction can be generalized as follows. Start
with a regular Sasaki manifold (M 2n+1, gM , η, ξ, Φ), and consider its Boothby-
Wang fibration p : M → N 2n. It is known (see [6]) that p is a circle bundle.
For an integer s, consider as before the iterated fibred product π : H2n+s → N ,
which becomes a torus bundle. The above argument goes through to prove
that π is a harmonic Riemannian submersion.

Acknowledgements. I am endebted to J.C. Wood for suggesting me this problem.
This work was partially supported by a TEMPUS grant and by the ANCS contract
2-CEx06-11-12/25.07.2006.

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Received 8 January 2007 “Dun˘area de Jos” University
Department of Mathematics

Str. Domneasc˘a 111
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