Rend. Istit. Mat. Univ. Trieste
Vol. XXX, 45{55 1998
Totally
Geo desic Horizontally
Conformal Maps
M.T. Mustafa
Summary. - We obtain a characterization of total ly geodesic hor-
izontal ly conformal maps by a method which arises as a conse-
quenceof the Bochner technique for harmonic morphisms. As a
geometric consequence we show that the existence of a non-con-
stant harmonic morphism from a compact Riemannian mani-
m
fold M of non-negative Ricci curvature to a compact Rieman-
m
nian manifold of non-positive scalar curvature, forces M either
to be a global Riemannian product of integral manifolds of vertical
and horizontal distributions or to becovered by a global Rieman-
nian product.
1. Intro duction
A smo oth map : M 7! N between Riemannian manifolds is called
a harmonic morphism if it preserves germs of harmonic functions,
i.e. if f is a real valued harmonic function on an op en set V
1
N then the comp osition f is harmonic on V M . Due
to a characterization obtained by B. Flugede [6] and T. Ishihara
[9 ], harmonic morphisms are precisely the harmonic maps which are
horizontally weakly conformal.
J. Vilms in [13 ] carried out a study of totally geo desic maps with
emphasis on totally geo desic Riemannian submersions and showed
Author's address: GIK Institute of Engg. Sciences and Technology, Topi,
Sabi, NWFP,Pakistan
46 M. T. MUSTAFA
that these can be characterized as Riemannian submersions with
totally geo desic bres and integrable horizontal distribution. This
pap er is aimed to achieve an analogous characterization of totally
geo desic horizontally conformal maps and to obtain geometric con-
sequences of this characterization in the study of totally geo desic
horizontally conformal maps. The metho d of pro of uses, as to ols,
the Weitzenbock formula for harmonic morphisms develop ed by the
author in [11 ] and the fundamental equations of horizontally confor-
mal submersions studied by S. Gudmundsson in [7 ].
The remaining part of this section presents a brief intro duction
to harmonic morphisms, the Bo chner technique for harmonic mor-
phisms and the fundamental equations of horizontally conformal sub-
mersions.
m n
1.1. Harmonic Morphisms. Recall that a map : M 7! N is
harmonic if and only if its tension eld = trace rd vanishes.
The reader is referred to [2 ], [3] and [4 ] for a comprehensive account
of harmonic maps.
m n
De nition 1.1. A map : M 7! N between Riemannian mani-
folds is called a harmonic morphism if f is a real valued harmonic
1
function on V M for every real valued function f which is
1
harmonic on an op en subset V of N with V non-empty.
m n
For a smo oth map : M 7! N , let C = fx 2 M j rank d <
x
ng b e its critical set . The p oints of the set M nC are called regular
V
M at x is de ned points. For each x 2 M nC , the vertical space T
x
V H
by T M = Ker d . The horizontal space T M at x is given by
x
x x
V V
the orthogonal complementofT M in T M so that T M = T M
x x
x x
H
T M .
x
m M n N
De nition 1.2. A smo oth map : M ; h; i ! N ; h; i
is called horizontal ly weakly conformal if d = 0 on C and the
restriction of to M nC is a conformal submersion, that is, for each
H
x 2 M nC , the di erential d : T M 7! T N is conformal and
x
x
x
+
surjective. This means that there exists a function : M nC ! R
such that
N
2 M H
h d X ;d Y i = hX; Y i ; 8X; Y 2 T M:
TOTALLY GEODESIC etc. 47
+
By setting = 0 on C , we can extend : M ! R to a
0
2 2
continuous function on M such that in smo oth, in fact =
+
2
kd k =n. The function : M ! R is called the dilation of the map
0
. Harmonic morphisms can b e characterized as follows.
m M n N
Theorem 1.3 [6],[9] . A map : M ; h; i ! N ; h; i is
a harmonic morphism if and only if it is a harmonic and horizontal ly
conformal map.
We refer the reader to [1 , 6, 14 ] for an intro duction and basic
prop erties of harmonic morphisms. For an up dated list of harmonic
morphisms bibliography, see [8 ].
In [11 ], the author develop ed a Bo chner technique for harmonic
morphisms and obtained the following Weitzenbock formula for har-
monic morphisms b etween Riemannian manifolds.
m n
Prop osition 1.4. Let M and N be Riemannian manifolds. Let
m n
: M ! N bea harmonic morphism with dilation . Then
n
2 2 4 N 2 M
= krd k + Scal Scal j ; 1
H
2
where
n
X
M
Ricci e ;e ; Scal j =
s s
H
s=1
denotes the Hodge-deRham Laplacian on functions on M and
n m H V
e , e are orthonormal bases of T M and T M respec-
s s
x x
s=1 s=n+1
m V
tively, so that e is an orthonormal basis of T M = T M
s x
x
s=1
H
T M .
x
1.2. Fundamental equations of horizontally conformal
submersions. The fundamental equations of horizontally confor-
mal submersions were established by S. Gudmundsson in [7] as a
generalization of the fundamental equations of submersions found
by O'Neill [12 ].
We recall that the fundamental tensors T , A of a submersion are
de ned as
T F = Hr V F + Vr H F;
E V E V E
A F = Hr V F + Vr H F;
E HE HE
48 M. T. MUSTAFA
m
where E; F are vector elds on M and H ; V denote the orthogonal
pro jections on the horizontal and vertical spaces resp ectively.
Notice that T restricted to vertical vector elds gives the second
fundamental form of the bres of the submersion and it can b e easily
seen that T = 0 is equivalent to the condition that the bres are
totally geo desic submanifolds.
For a horizontally conformal submersion it was shown that the
following relation holds.
m M n N
Prop osition 1.5. Let : M ; h; i ! N ; h; i be a hori-
zontal ly conformal submersion with dilation and X; Y be horizontal
vectors, then
o n
1 1
2
A Y = : V [X; Y ] grad
X V
2
2
From the ab ove prop osition it is obvious that in case of vertically
homothetic horizontally conformal submersion, the tensor A of the
horizontally conformal submersion reduces to the tensor A of a Rie-
mannian submersion, i.e. it b ecomes the integrability tensor of the
horizontal distribution. On the other hand we see that the vanishing
of A identically, implies that the horizontally conformal submersion
is vertically homothetic.
Here we only state the curvature equations relevant to our work
and refer the reader to [7 ] for other fundamental curvature equations
of horizontally conformal submersions.
m M n N
Lemma 1.6. Let m>n 2 and : M ; h; i ! N ; h; i
+
be a horizontal ly conformal submersion, with dilation : M ! R .
If X; Y are horizontal and U; V are vertical vectors, then
M M
M M
hR X; U X; U i = h r A X; U i + hA U; A U i +
U X X X
M
M
h r T X; U i hT X; T X i +
X U U U
E D
M
1
2 M
: + hA X; U i U; grad
X V
2
1 3
M N 2
M N
~ ~ ~ ~
hR X; Y X; Y i = hR X; Y Y;X i kV [X; Y ]k +
2
4
h D E D E
2
M M
1 1
M M
+ hX; Y i r grad ;X hY; Y i r grad ;X +
Y X
2 2
2
TOTALLY GEODESIC etc. 49
D D E E i
M M
1 1
M M
+hY; X i r grad hX; X i r grad ;Y ;Y +
X Y
2 2
h i
4
2 2
1 1 1
2
Y Y X kX ^ Y k grad + : X +
2 2 2
4
2. Totally geo desic horizontally conformal maps
m n
De nition 2.1. A map : M ! N is total ly geodesic if and only
if its second fundamental form rd vanishes, where
1
TN
M
rd X; Y = r d Y = r d Y d r Y ;
X
X
X
for X; Y 2C TM .
These maps are characterized as the maps which take geo desics
m n
of M linearly to geo desics of N .
Lemma 2.2. A total ly geodesic map has constant rank and constant
1
2
kd k . In particular, a total ly energy density e , where e =
2
geodesic horizontal ly conformal map has constant dilation.
Proof. cf. [5 , p. 15].
In this section we develop a relation b etween the second funda-
mental form and the fundamental tensors A and T of a horizontally
conformal submersion, in order to achieve the following characteri-
zation of totally geo desic horizontally conformal maps b etween Rie-
mannian manifolds.
Theorem 2.3. Characterization of totally geo desic horizontally
m M n
conformal maps. Let m > n 2 and : M ; h; i ! N ;
N
h; i be a horizontal ly conformal map. Then is total ly geodesic
if and only if has constant dilation, total ly geodesic bres and in-
tegrable horizontal distribution.
Before proving Theorem 2.3 we will prove a few results, needed
in the pro of of Theorem 2.3. A necessary curvature relation b etween
M N
m
Scal j and Scal , for a horizontally conformal map : M !
H
n
N , is given by the following.
50 M. T. MUSTAFA
m n
Prop osition 2.4. Let m > n 2 and M , N be Riemannian
m M n N
manifolds. Let : M ; h; i ! N ; h; i be a horizontal ly
+
conformal submersion, with dilation : M ! R . Then
n
X
3
2
M 2 N
kV [e ;e ]k + Scal j = Scal +
t s
H
4
s;t=1
E D E i h D
2
M M
1 1
grad grad r + r + ;e ;e
e e t s
s t
2 2
2
i h
4
2 2
1 1 1
+ + e e + grad
t s
2 2 2
4
n m
n o
X X
M
2 2
+ kA e k h r T e ;e i kT e k +
e t e e s t e s
s s t t
s=1 t=n+1
m
D E hD E i
2 2
X
M M 2
1 1
r grad ;e grad ;e ; +
e V t V t
t
2 2
2 2
t=n+1
where
n
X
M
Scal j = Ricci e ;e ;
s s
H
s=1
m n m
for an orthonormal basis e of T M such that e , e
s x s s
s=1 s=1 s=n+1
V H
M respectively. M and T are orthonormal bases of T
x x
M
Proof. We can write Scal j as
H
n
X
M
Scal j = Ricci e ;e 2
s s
H
s=1
m n n
X X X
M M
M M
h R e ;e e ;e i : h R e ;e e ;e i + =
t s t s t s t s
t=n+1 s=1 s;t=1
Computing the right hand side using Lemma 1.6 gives the required
relation.
The following result on totally geo desic submersion will b e needed
later.
m M n N
Lemma 2.5. Let : M ; h; i ! N ; h; i be a submersion.
If is total ly geodesic then the bres are total ly geodesic.
TOTALLY GEODESIC etc. 51
1
Proof. Let F = x b e the bre at x 2 M . Let i : F 7! M be
x x
V
the inclusion map, then i is constant. Therefore, for U; V 2 T M
x
rd i U; V =0
d rdi U; V = rd diU; diV =0
rdi U; V =0:
Hence the bres are totally geo desic.
The integrability of the horizontal distribution of a totally geo de-
sic horizontally conformal map is achieved by combining Prop osi-
tion 2.4 and the Weitzenbock formula for harmonic morphisms. Pre-
cisely,wehave
m M n N
Lemma 2.6. Let m>n 2 and : M ; h; i ! N ; h; i
be a total ly geodesic horizontal ly conformal map. Then the horizontal
distribution is integrable.
Proof. Knowing that a totally geo desic horizontally conformal map
has totally geo desic bres and constant dilation, wehave from Prop o-
sition 2.4
n n m
X X X
3
2
2 M 2 N
kV [e ;e ]k + kA e k : Scal j = Scal
t s e t
s
H
4
s;t=1 s=1 t=n+1
Using in Weitzenbock formula for harmonic morphisms wehave
m n n
X X X
3
2
2
kA e k ; kV [e ;e ]k =
e t t s
s
4
t=n+1 s=1 s;t=1
or we can write as
n n m
X X X
2 2
3 kA e k = kA e k : 3
e s e t
t s
s;t=1 s=1 t=n+1
But we know that
n n m
X X X
2 2
kA e k = kA e k ; 4
e s e t
t s
s;t=1 s=1 t=n+1
52 M. T. MUSTAFA
as follows.
n n m
X X X
2
kA e k = hA e ;e ihA e ;e i
e s e s u e s u
t t t
s;t=1 s;t=1 u=n+1
n m
X X
= hA e ;e ihA e ;e i
e u s e u s
t t
s;t=1 u=n+1
n m
X X
2
= kA e k :
e u
t
t=1 u=n+1
Comparing Equations 3 and 4 , wehave
n
X
2
kA e k =0:
e s
t
s;t=1
Hence, the horizontal distribution is integrable.
Proof of Theorem 2.3.
= Follows from Lemma 2.5, Lemma 2.6 and Lemma 2.2.
= If has totally geo desic bres, integrable horizontal distri-
bution and constant dilation then it follows from Prop osition 2.4
that
M 2 N
Scal j = Scal : 5
H
Substituting in Weitzenbock formula for harmonic morphism we ob-
tain rd =0.
Having seen that a totally geo desic horizontally conformal map
has integrable horizontal distribution, we consider the horizontal fo-
m
liation on M and obtain the geometric consequences of the ab ove
characterization on the horizontal and vertical foliations.
m M n N
Theorem 2.7. Let : M ; h; i ! N ; h; i be a total ly
geodesic horizontal ly conformal map between Riemannian manifolds
with m>n 2. Then
m
1: The horizontal foliation is total ly geodesic in M .
2: The vertical foliation is Riemannian with bund le like metric.
TOTALLY GEODESIC etc. 53
Proof. 1. From Theorem 2.3, the horizontal distribution is inte-
grable and the dilation is constant, therefore, wehave a horizontal
m
foliation on M which is totally geo desic as follows.
Let X; Y b e horizontal vectors, then Prop osition 1.5 implies that
M
Vr Y = A Y =0: 6
X
X
Let denote the second fundamental form of a leaf L of the hori-
L
zontal foliation, then
M
X; Y =Vr Y =0: 7
L
X
Hence every leaf of horizontal foliation is a totally geo desic subman-
ifold of M .
2. It is shown in [14 ] that 1 = 2 .
Remark 2.8. Since a totally geo desic horizontally conformal map
has totally geo desic bres, therefore, switching over the role of ver-
tical and horizontal foliations in Theorem 2.7 we observe that in
case of a totally geo desics horizontally conformal map, the horizon-
tal foliation is also Riemannian. Hence, we conclude that a totally
geo desic horizontally conformal map gives rise to orthogonal folia-
m
tions on M , namely horizontal and vertical, which are Riemannian
with totally geo desic leaves.
A further application of Theorem 2.3 yields the following result re-
garding existence of totally geo desic harmonic morphisms.
m n
Lemma 2.9. A harmonic morphism : M ! N is total ly geodesic
if and only if ker d is holonomy invariant and has constant dila-
tion.
Proof. The only if part follows from Lemma 2.2 and [13 , p. 74].
Conversely supp ose that ker d is holonomy invariant. Then
the horizontal and vertical distributions are integrable with totally
geo desic integral manifolds cf. [10 , pp. 181{182]. The result then
follows from Theorem 2.3.
The ab ove analysis combined with the applications of the Bo chner
technique presented in [11 ] lead to the following decomp osition result
for a compact Riemannian manifold of non-negative Ricci curvature, admitting a harmonic morphism.
54 M. T. MUSTAFA
m M
Theorem 2.10. Let M ; h; i be a compact Riemannian man-
M
n N
ifold with Ricci 0 and N ; h; i be a Riemannian manifold
N
m n
with Scal 0. Let be a harmonic morphism from M to N .
m
Then either M is a global Riemannian product or it is covered by
a global Riemannian product.
m
Proof. By [11 , Theorem 2.5] is totally geo desic. If M is simply-
connected then pro of follows from Theorem 2.7, Lemma 2.9 and [10 ,
p. 187].
m
~
Supp ose M is non-simply-connected. Let M be its universal
covering space. Since can b e lifted to a totally geo desic horizontally
~
~
conformal map from M , therefore, the pro of follows by rep eating
the ab ove argument.
Acknowledgments. The author is grateful to John C. Wood and
S. Gudmundsson for imp ortant comments and useful hints during
the completion of this work.
References
[1] Baird P., Harmonic maps with symmetry, harmonic morphisms, and
deformation of metrics, Pitman Res. Notes Math. Ser. 87, Pitman,
Boston, London, Melb ourne, 1983.
[2] Eells J. and Lemaire L., Areport on harmonic maps, Bull. London
Math. So c. 10 1978 , 1{68.
[3] Eells J. and Lemaire L., Selected topics in harmonic maps,
C.B.M.S. Regional Conf. Series 50, AMS, 1983.
[4] Eells J. and Lemaire L., Another report on harmonic maps, Bull.
London Math. So c. 20 1988 , 385{524.
[5] Eells J. and Ratto A., Harmonic maps and minimal immersions
with symmetries, Ann. Math. Stud. 130, Princeton University Press,
1993.
[6] Fuglede B., Harmonic morphisms between Riemannian manifolds,
Ann. Inst. Fourier Grenoble 28 1978 , 107{144.
[7] Gudmundsson S., The geometry of harmonic morphisms, Ph.D. the-
sis, University of Leeds, 1992.
[8] Gudmundsson S., Bibliography of harmonic morphisms,
http://www.maths.lth.se/matematiklu/p ersonal/sigma/harmonic/
bibliography.html.
TOTALLY GEODESIC etc. 55
[9] Ishihara T., A mapping of Riemannian manifolds which preserves
harmonic functions, J. Math. Kyoto Univ. 19 1979 , 215{229.
[10] Kobayashi S. and Nomizu K., \The foundations of di erential ge-
ometry I", Wiley Interscience, New York, 1963.
[11] Mustafa M.T., A Bochner technique for harmonic morphisms, J.
London Math. So c., to app ear.
[12] O'Neill B., The fundamental equations for a submersion, Michigan
Math. J. 13 1966 , 459{469.
[13] Vilms J., Total ly geodesic maps, J. Di erential Geometry 4 1970 ,
73{79.
[14] Wood J.C., Harmonic morphisms, foliations and Gauss maps, Com-
plex di erential geometry and nonlinear partial di erential equations
Providence, R.I. Y.T. Siu, ed. , Contemp. Math. 49, Amer. Math.
So c., Providence, R.I., 1986, pp. 145{184.
Received June 6, 1997.