Rev.Acad.Canar.Cienc., IX (Núm. 1) , 131-139 (1997)
A SCHUR-LIKE THEOREM FOR NORMAL FLOW SPACE
FORMS
J. C. González-Dávila*
Departamento de Matemática Fundamental
Sección de Geometría y Topología
Universidad de La Laguna
La Laguna, Spain
ABSTRACT
L. Vanhecke
Department of Mathematics
Katholieke Universiteit Leuven
Celestijnenlaan 200 B
3001 Leuven, Belgium
We prove a Schur-like theorem for normal flow space forms and extend sorne
known characterization theorems for ·this class of spaces.
1991 Mathematics Subject Classification. 53C25.
Key words and Phrases. Flows, normal and contact flows, normal flow space forms, Schur-like
theorem.
l. INTRODUCTION
It is a well-known fact that the sectional curvature function on a Riemannian manifold
of dimension > 2 determines the whole Riemann curvature tensor. The theorem of
Schur states that this sectional curvature function is globally constant if it is pointwise
constant. In that case the manifold is called a real space form. Severa! similar results
have been proved in the framework of other geometries by restricting the sectional curvature
function to special families of planes naturally defined by the given geometrical
structure. Holomorphic, quaternionic and <,P-holomorphic sectional curvature functions
are well-known in Kiihler, quaternionic Kiihler and Sasaki geometry. Complex, quaternionic
Kiihlerian and Sasakian space forros are then defined accordingly. Severa! other
examples could be given, also for pseudo-Riemannian geometry.
In a series of papers, we have generalized Sasakian geometry by considering the geometry
on a Riemannian manifold which is equipped with a Riemannian flow generated
'Supported by the Consejería de Educación del Gobierno de Canarias.
l :l l
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by a unit Killing vector field. We called this fiow geometry. (See (2]: (3] for the basic
material.) The presence of families of special planes gave rise to the introduction of the
notion of a fiow space Jorm and a first study of these spaces, in particular for normal
flow space forms, has been developed in (4]. There it was shown that fundamental
differences appear when compared with the study of Sasakian space forms.
The main purpose of this paper is to prove a Schur-like theorem for this class of
space forms . At the same time we obtain sorne improvements of theorems, proved in
[4], [5], and which are related to the characterization of these space forms.
2. PRELIMINARIES
Let (M,g) be an n-dimensional, connected, smooth Riemannian manifold with
n ~ 2. Furthermore, Jet V denote its Levi Civita connection and R the corresponding
curvature tensor determined by Ruv = V¡u,V] - [Vu, Vv] for U, V E X(M) , the Lie
algebra of smooth vector fields on M. Next, Jet (M, g) be equipped with an isometric
fiow [9] 3'( generated by a unit Killing vector field (. Vectors which are orthogonal to
( are called horizontal vectors.
Now, put HU= -Vu( and h(U, V)= g(HU, V) for ali U, V E X(M) . Since ( is
a Killing vector field, h is skew-symmetric. Moreover, h = -dr¡ where r¡ is the metric
dual one-form of (. Further, we have
(1) R(X, ( ,Y,() = g(HX,HY) = -g(H2X,Y) .
This implies that the (-sectional curvature K(X, O is non-negative. It is strictly positive
when H is of maximal rank. Since H( = O, in that case the rank is n - 1 or
equivalently, r¡ is a contact form. Such a flow is called a contact fiow.
The flow 3'( determines locally a Riemannian submersion. In fact, for each point
m in M, Jet U be an open neighborhood of m such that ( is regular on U. Then, the
mapping rr : U -t U = U/( is a submersion. Furthermore, !et g denote the induced
metric on U defined by
?J(X, Y)*= 9(x·, Y*)
for X, y E X(U.) and where x·, y• denote the horizontal lifts of X, y with respect to
the distribution on U determined by r¡ = O. Then 1r: (U,g¡u) -t (U,g) is a Riemannian
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submersion. The Levi Civita connection V on (U,g) is determined by
(2) v g.Y* = (V gY)* + h(X*, Y*)e
for X, Y E :r(U).
Next, the flow Je is said to be normal if, for ali horizontal vectors X, Y , the curvature
transformations Rxy leave the horizontal subspaces invariant, or equivalently,
R(X, Y, X , O = O. Then we have [2]
(3) {
Ruve = r¡(V)H2U - r¡(U)H2V,
Ru{V = g(HU, HV)e + r¡(V)H2U
for ali U, V E :r( M). Using (2), it follows that the curvature tensors of V and V are
related by
(4)
for X, Y, Z E :r(U).
Rx+Z* ._ g(HY*, Z*)H x· + g(HX*, Z*)HY*
+2g(HX*, Y*)HZ*
The following lemma plays a crucial role.
Lemma 2.1. (6], [7] Let ( M, g) be a Riemannian manifold equipped with a normal
flow Je- Then the eigenvalues of the symmetric endomorphism field H2 are constant.
Let '.K denote the horizontal distribution on (M,g, Je ) and denote by -cf, ... , -e;
the different constant eigenvalues of Hin and by '.Ket ( m ), i = 1, ... , r, the corresponding
eigenspaces for each m E M. Then, using (1), we get
'.Ket (m) = {X E '.K(m) 1 K(X,e) = e;}.
Furthermore, '.}{et, i = 1, ... , r, determines a differentiable distribution on M and for
each m E M, '.K( m) = '.J{ef ( m) ffi ... ffi '.Ke~ ( m) is an H-invariant orthogonal decomposition
of '.K(m).
We also obtain distributions '.}{et, i = 1, .. . , r, on each base space U of the local fibration
which assign to each point m = rr( m) of U the subspac:es '.Ke; ( m) = rr •m'.Ke; ( m)
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of TmÜ. Note that '.]{et is obtained locally as the horizontal lift of '.ke;. These distributions
are differentiable and involutive and furthermore, Ü can be expressed as the
Riemannian product Ü = Ü1 x ... x Ür , where Ü; ( i = 1, ... , r) is the maximal integral
manifold of '.ke; through a point of Ü. See [7] for more details.
Furthermore, for normal contact flows we have
Lemma 2. 2. [7] A normal contact fiow on a Riemannian manifold ( M , g) fibers
local/y ove, a Kahler manifold. Moreover, if Ü = Ü1 x ... x Ür is a base space o/ a local
fibration, then
-1 - -1 - J = C1 H1 X ... X cr Hr
is a Hermitian structure on (Ü,g). Here ÍI; = Íl •p¡ where p¡ denotes the projection o/
Ü onto Ü; .
This implies that for normal contact flows we have dim M 2n + 1, n
dim'.Ket = 2n;, i = 1, ... , ,.
3. THE SCHUR-LIKE THEOREM AND SOME NEW RESULTS
First, we start with sorne considerations about the typical sectional curvatures for
(M, g, J~ ). We considered already the ~-sectional curvature and introduce now the Hsectional
curvature. A plane section in TmM, m E M, is called an H-section if there
exists a horizontal vector X E T m M such that { X, H X} is a basis of the plane section.
The corresponding sectional cnrvature K(X, H X) is called the H-sectional curvature
determined by X. An H-section on (U, g) determines an Íl-section on the base space
(Ü,g) of the local fibration rr : U-+ Ü = U/( Using (1), (4), we then obtain at once
the following relation between the ~-, H- and Íl-sectional curvature:
(5) (k(x, ÍIX))* = K(X*, H x·) + 3K(X*, O
where X E X(Ü).
r
Now, !et X = ¿ X; be a horizontal vector field on (M, g, J~) where X; E '.]{et for
i=l
i = 1, ... , ,. Then we have
(6) 1 ~ 2 2
K(X, O = IIXll2 ~ c¡ IIX;II
l :S4
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and hence, we obtain
Proposition 3.1. Let J( be a normal fiow on ( M, g ). Then the (-sectional curvatvres
are complete/y determined by the scalars c;, i = 1, ... , r.
Furthermore, by using Lemma 2.2 and following the proof of [4, Theorem 3.1], we
get the following extension of that Theorem 3.1:
Theorem 3.1. On a Riemannian manifold equipped with a normal contact fiow,
the H - and (-sectional curvatures determine the curvature of ( M, g) complete/y.
Now, we turn to the consideration of the normal flow space forms and the corresponding
Schur-like theorem. A Riemannian manifold ( M, g) equipped with a contact
flow J( is said to be a fiow space form if the H-sectional curvature is pointwise constant,
that is, K(X, H X) is independent of the horizontal vector X E TmM at each
point m E M. These space forms have been üitroduced in [4] where two cases are considered
according to whether the (-sectional curvature is (pointwise) constant or not.
When the (-sectional curvature is (pointwise) constant, each normal flow space form
is obtained by a homothetic change of metric from a Sasakian space forro . Hence, in
this case and for dim M ~ 5, if the H-sectional curvature is pointwise constant, then
it is globally constant [l]. This proves the Schur-like theorem for that case.
Next, we consider the second case and use Lemma 2.2 to prove the required theorem.
Theorem 3.2. Let (M,g,J() be a (2n + 1)-dimensional normal fiow space form
of pointwise constant H -sectional curvature k and with non-constant (-sectional curvature.
Then k is global/y constant. Moreover, '.}{2¡:J{ has exactly two eigenvalues -e¡, -e~,
( c1 c2 -/= O), and k is a strictly negative constant given by
r
Proof. Let X be a horizontal vector field and put X=¿ X;, X; E '.}{e~ . Using (5) and
i=l
Lemma 2.2, we obtain
(7) K(Xi,JX;) = K(Xi, HX;) = k + 3cf , i = 1, ... ,r.
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So, we get
(8)
r
¿(k + 3cncl11X;ll 4
k(X, iI X) = --1=-1-------(
t IIXdl2) ( t c;IIXill 2 )
i=D j=I
Using this and (6), we rewrite (5) in the form
and hence, we obtain
r L cf(k + 3c;)IIX;ll 2 IIXill2 = O
i,J=l
i:¡l;
r I:: cfc;11xi11 2 11xj112
i ,J=l
k = -3 ... ~ < o.
L cfllX;ll 2 IIXill2
i ,J=l
i:¡l;
Now, put X = Xh + X1, 1 ::; h-/- l::; r. Then the expression for k gives
If r ~ 3, we obtain el = c2 for i = 1, ... , r, and we get a contradiction. So r = 2 and
this completes the proof. •
It follows from (7) that each base space Ü = Ü1 x Ü2 of a normal flow space form
with non-constant (-sectional curvature is a product of Kiihler manifolds of constant
holomorphic sectional curvatures h¡ = k + 3cr, i = l , 2. This implies that the manifold
(M, g, J() is locally transversally modelled on cpn, (h1 ) x cHn, (h2 ) if e? > e~. From this
consideration and using the proof of [4, Theorem 4.2], we obtain at once the following
improvement of that theorem.
Theorem 3.3. Let J( be a normal contact flow on a (2n + 1 )-dimensional Riemannian
manifold (M, g) with non-constant (-sectional curvature. Then (M, g, J() is
a fiow space form if and only if it is transversally modelled on cPn1 (h1) x cHn2 (h2)
where 1h2 1 < h1, n1 + n 2 = n , and the (-sectional curvatures cf, i = l , 2, satis/y
(9)
Furthermore, the H-sectional curvature k is given by
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and the curvature tensor is given by
¡3(h1+h2)( . ) + (-1) 4(hi _ h2) g(HY; , Z;)HX;-g(HX;, Z;)HY;- 2g(HX;, Y;)HZ;
+ (-1) ¡h;(h1-h2){( ) 3(hi + h2) g(Y;, Z;)r¡(U) - g(X;, Z;)r¡(V) (
+r¡(W)(r¡(V)X; - r¡(U)Y;)}}
+g(HV, W)HU - g(HU, W)HV - 2g(HU, V)HW
2 2 2
for vector fields U=¿ X;+ r¡(U)(, V=¿ Y;+ r¡(V)(, W = ¿ Z; + r¡(W)( on M.
i=l i=l i=l
Finally, we derive a criterion for (M,g,Je) to be a normal flow space form of the
second kind. It improves a result derived in [5] .
Theorem 3.4. Let Je be a normal contact fiow on a (2n + ! )-dimensional Riemannian
manifold ( M , g) such that H12.x_ has exactly two eigenvalues -cL -e~ and
dim '.Kc~ = 2n; ~ 4 / or i = 1, 2. Then ( M, g, J () is a fiow space f orm ( with non-constant
e-sectional curvature) if and only if
are proportional to H X for ali X = X 1 + X 2 E '.Kc¡ EB '.}{e~ and where H .LX denotes a
vector orthogonal to HX in the plane spanned by HX1 and HX2 when IIXill llX2II /:- O.
Proof. First, let (M , g,Je) be such a normal flow space form. Then the result follows
directly from the expression for the curvature tensor given in Theorem 3.3.
Conversely, if 11xi11-2Rx,Hx,X1 + 11x211-2Rx2HX2X2 = kHX for X E TmM, that
is, Rx,Hx;X; = kllX;ll2HX;, i = 1, 2, then using (4) and Lemma 2.2, we have, for each
uni t vector X;,
Now let c1 > c2 > O. Then, it follows from this relation and [8, Theorem AJ that
(M,g, Je ) is transversally modelled on a Riemannian product M = M1 x M2 where
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M; is a complex space form of constant holomorphic sectional curvature h; = k + 3cf,
i = 1,2. From (3), (4) and the expression for the curvature tensor of a complex space
form (see for example [10]), we get
+3g(HY,X)HX,
where X= X1 + X2 , Y= Y1 +},; and X;, Y; E '.Kc7(m). So, RxH1.xX is proportional
to
and hence, it is proportional to H X if and only if
This, together with the expressions for h;, implies that h1 > O and h2 < O.So, (M, g, 3"~)
is transversally modelled on cPn1 (h¡) x cHn2 (h2 ) where 1h21 < h1 , n 1 + n2 = n.
Moreover, it is easy to check that (9) is satisfied. So, the result follows by applying
Theorem 3.3. •
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