A NOTE ON GEODESIC MAPPINGS OF PSEUDOSYMMETRIC

A NOTE ON GEODESIC MAPPINGS OF PSEUDOSYMMETRIC

COLLOQUIUM MATHEMATICUM
VOL. LXII
1991
FASC. 2
A NOTE ON GEODESIC MAPPINGS
OF PSEUDOSYMMETRIC RIEMANNIAN MANIFOLDS
BY
FILIP D E F E V E R * (LEUVEN)
AND
RYSZARD D E S Z C Z (WROCLAW)
1. Introduction. Let (M, g) be a connected n-dimensional , n ≥ 3,
e R, S and κ the
semi-Riemannian smooth manifold. We denote by ∇, R,
Levi-Cività connection, the curvature tensor, the Riemann–Christoffel curvature tensor, the Ricci tensor and the scalar curvature of (M, g), respectively. We define on M the tensor fields R.R and Q(g, R) by the formulas
(R.R)(X1 , X2 , X3 , X4 ; X, Y )
e
e
= − R(R(X,
Y )X1 , X2 , X3 , X4 ) − R(X1 , R(X,
Y )X2 , X3 , X4 )
e
e
− R(X1 , X2 , R(X, Y )X3 , X4 ) − R(X1 , X2 , X3 , R(X,
Y )X4 ) ,
Q(g, R)(X1 , X2 , X3 , X4 ; X, Y )
= R((X ∧ Y )X1 , X2 , X3 , X4 ) + R(X1 , (X ∧ Y )X2 , X3 , X4 )
+ R(X1 , X2 , (X ∧ Y )X3 , X4 ) + R(X1 , X2 , X3 , (X ∧ Y )X4 ) ,
where
(X ∧ Y )Z = g(Y, Z)X − g(X, Z)Y ,
X, Y, Z, X1 , . . . , X4 ∈ Ξ(M ), Ξ(M ) being the Lie algebra of vector fields
on M .
A semi-Riemannian manifold (M, g) is said to be pseudosymmetric ([4])
if at every point of M the following condition is satisfied:
(∗)
the tensors R.R and Q(g, R) are linearly dependent.
A semi-Riemannian manifold (M, g) is pseudosymmetric if and only if
(1)
R.R = LQ(g, R)
on the set UR = {x ∈ M | Z(R) 6= 0 at x}, where L is a function on UR and
κ
G
Z(R) = R −
n(n − 1)
*Aspirant NFWO, Belgium.
314
F. D E F E V E R AND M. D E S Z C Z
with G defined by
G(X1 , X2 , X3 , X4 ) = g((X1 ∧ X2 )X3 , X4 ) ,
X1 , . . . , X4 ∈ Ξ(M ).
If R.R = 0 on M , then the manifold (M, g) is called semisymmetric ([9]).
The local and global structure of Riemannian semisymmetric manifolds was
described in [9] and [10]. The class of pseudosymmetric manifolds is essentially wider than the class of semisymmetric manifolds (cf. [3], [4], [2], [7]).
The study of totally umbilical submanifolds of semisymmetric manifolds as
well as the consideration of geodesic mappings onto semisymmetric manifolds lead to the concept of pseudosymmetric manifolds (see [1], [5], [6],
[8]).
In the survey paper [8] the following theorem is presented.
Theorem 1.1 ([8], Theorem 3). If (M, g) is a pseudosymmetric semiRiemannian manifold admitting a non-trivial geodesic mapping f onto a
manifold (M , g) then (M , g) is also a pseudosymmetric manifold.
Unfortunately, the proof of this theorem is not published. On the other
hand, Theorem 1.1 is very important in the study of pseudosymmetric manifolds. In this paper we give a proof of this theorem.
2. Preliminaries. Let (M, g), n = dim M ≥ 3, be a semi-Riemannian
manifold covered by a system of coordinate neighbourhoods {V ; xj }. We
h
denote by Γijh , gij , Rijk
, Rhijk and Sij the local components of the Levie R and
Cività connection ∇ and the local components of the tensors g, R,
S, respectively. Further, we denote by
(2)
(R.R)hijklm = ∇m ∇l Rhijk − ∇l ∇m Rhijk
(3)
= −Rrijk Rrhlm − Rhrjk Rrilm − Rhirk Rrjlm − Rhijr Rrklm ,
Q(g, R)hijklm = ghm Rlijk + gim Rhljk + gjm Rhilk + gkm Rhijl
− ghl Rmijk − gil Rhmjk − gjl Rhimk − gkl Rhijm ,
the local components of the tensors R.R and Q(g, R), respectively.
For a (0, 2)-tensor field A on (M, g) one defines the endomorphism X∧A Y
of Ξ(M ) by the formula
(X ∧A Y )Z = A(Y, Z)X − A(X, Z)Y
where X, Y, Z ∈ Ξ(M ). In particular, we have
X ∧g Y = X ∧ Y .
Further, for a (0, k)-tensor field T on (M, g), k ≥ 1, we define the tensor
field Q(A, T ) by the formula
Q(A, T )(X1 , . . . , Xk ; X, Y ) = T ((X ∧A Y )X1 , X2 , . . . , Xk )
PSEUDOSYMMETRIC RIEMANNIAN MANIFOLDS
315
+ T (X1 , (X ∧A Y )X2 , . . . , Xk ) + . . . + T (X1 , . . . , Xk−1 , (X ∧A Y )Xk ) ,
where X, Y, X1 , . . . , Xk ∈ Ξ(M ). Evidently, putting in the above formula
A = g, T = R we obtain the tensor field Q(g, R).
Let (M, g) and (M , g) be two n-dimensional semi-Riemannian manifolds.
A diffeomorphism f : M → M which maps geodesic lines into geodesic lines
is called a geodesic mapping. It is known that in a common coordinate
system {x1 , . . . , xn }, Christoffel symbols and curvature tensors of (M, g)
and (M , g) are related by
Γ hij = Γijh + δih ψj + δjh ψi ,
(4)
Rhijk = Rhijk + δjh ψik − δkh ψij ,
(5)
where
ψij = ∇j ψi − ψi ψj ,
det g 1
∂
.
ψi =
log 2(n + 1) ∂xi
det g (6)
(7)
In the sequel such a geodesic mapping of (M, g) onto (M , g) will be
ψ
denoted by f : (M, g)−→(M , g) and the manifolds (M, g) and (M , g) will
ψ
be called geodesically related. A geodesic mapping f : (M, g)−→(M , g) is
called non-trivial on M if the covector field ψ with the local components ψi
is non-zero.
ψ
R e m a r k. If f : (M, g)−→(M , g) is a geodesic mapping, then f (UR ) =
UR . We can prove this using the fact that the Weyl projective curvature
tensor W , defined by
1
e
(S(Y, Z)X − S(X, Z)Y ) ,
W (X, Y )Z = R(X,
Y )Z −
n−1
X, Y, Z ∈ Ξ(M ), is invariant under geodesic mappings and that W vanishes
at a point of M if and only if Z(R) vanishes at this point.
ψ
Lemma 2.1. Let f : (M, g)−→(M , g) be a geodesic mapping of a pseudosymmetric manifold (M, g) onto a manifold (M , g) and let the condition
R.R = LQ(g, R) be satisfied on UR . Then in a common coordinate system
{x1 , . . . , xn } on UR and UR
1
η(g hl Rmijk − g hm Rlijk )
n
+ Bil Rhmjk − Bim Rhljk + Bjl Rhimk − Bjm Rhilk + Blk Rhijm − Bkm Rhijl
(8)
(R.R)hijklm =
+ Fil Ghmjk − Fim Ghljk − Fjl Ghimk − Fjm Ghilk + Flk Ghijm − Fkm Ghijl ,
where
Bij = −Lgij + ψij ,
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F. D E F E V E R AND M. D E S Z C Z
1
1
Fij = − Aij − Lg rs (ψrs gij − grs ψij ),
n
n
Aij = ψir S rj − ψrs Rrij s ,
η = g rs (−grs L + ψrs ) .
P r o o f. Using the Ricci identity and (5) we obtain
∇m ∇l Rsijk − ∇l ∇m Rsijk = ∇m ∇l Rsijk − ∇l ∇m Rsijk
+ ψil Rsmjk − ψim Rsljk + ψjl Rsimk − ψjm Rsilk + ψkl Rsijm − ψkm Rsijl
+ δks (ψir Rrjlm + ψjr Rrilm ) − δjs (ψir Rrklm + ψkr Rrilm )
s
+ δls ψmr Rrijk − δm
ψlr Rrijk ,
which, by making use of (5), (R.R)sijklm = LQ(g, R)sijklm and (3), turns
into
h
∇m ∇l Rhijk − ∇l ∇m Rhijk = −L(δlh Rmijk − δm
Rlijk )
h
+ δlh Emijk − δm
Elijk + δkh (Eijlm + Ejilm ) − δjh (Eiklm + Ekilm )
+ ψil Rhmjk − ψim Rhljk + ψjl Rhimk − ψjm Rhilk + ψkl Rhijm − ψkm Rhijl
− L(gil (Rhmjk + δkh ψmj − δjh ψmk ) − gim (Rhljk + δkh ψlj − δjh ψlk )
h
+ gjl (Rhimk + δkh ψim − δm
ψik ) − gjm (Rhilk + δkh ψil − δlh ψik )
h
+ gkl (Rhijm + δm
ψij − δjh ψim ) − gkm (Rhijl + δlh ψij − δjh ψil )) ,
where
Emijk = ψmr R
But this, by contraction with g hs , gives
(9)
r
ijk
.
(R.R)hijklm = −L(g hl Rmijk − g hm Rlijk )
+ g hl Emijk − g hm Elijk + g hk (Eijlm + Ejilm ) − g hj (Eiklm + Ekilm )
+ ψil Rhmjk − ψim Rhljk + ψjl Rhimk − ψjm Rhilk + ψkl Rhijm − ψkm Rhijl
− L(gil (Rhmjk + g hk ψmj − g hj ψmk ) − gim (Rhljk + g hk ψlj − g hj ψlk )
+ gjl (Rhimk + g hk ψim − g hm ψik ) − gjm (Rhilk + g hk ψil − g hl ψik )
+ gkl (Rhijm + g hm ψij − g hj ψim ) − gkm (Rhijl + g hl ψij − g hj ψil )) .
Symmetrizing (9) in h, i we obtain
g hl Emijk − g hm Elijk + g il Emhjk − g im Elhjk
+ g hk (Eijlm + Ejilm ) + g ik (Ehjlm + Ejhlm )
− g hj (Eiklm + Ekilm ) − g ij (Ehklm + Ekhlm )
PSEUDOSYMMETRIC RIEMANNIAN MANIFOLDS
317
+ ψil Rhmjk − ψim Rhljk + ψhl Rimjk − ψhm Riljk
− L(g il Rmhjk + g hl Rmijk − g hm Rlijk − g im Rlhjk )
− L(gil (Rhmjk + g hk ψmj − g hj ψmk ) + ghl (Rimjk + g ik ψmj − g ij ψmk )
− gim (Rhljk + g hk ψlj − g hj ψlk ) − ghm (Riljk + g ik ψlj − g ij ψlk )
+ gjl (g hk ψim − g im ψhk + g ik ψhm − g hm ψik )
− gjm (g hk ψil − g il ψhk + g ik ψhl − g hl ψik )
+ gkl (g hm ψij − g ij ψhm + g im ψhj − g hj ψim )
− gkm (g hl ψij − g ij ψhl + g il ψhj − g hj ψil )) = 0 .
Contracting this with g hl and using the identity
(R.g)imjk = − gis Rsmjk − gms Rsijk = −gis Rsmjk − gms Rsijk
− gis (δjs ψmk − δks ψmj ) − gms (δjs ψik − δks ψij )
we get
(10) (n + 1)Emijk + Ejikm + Ekimj
+ g ik Ajm − g ij Akm + ηRimjk − nLRmijk
− L(n(gjm ψik − gkm ψij ) + g rs grs (g ik ψmj − g ij ψmk )
+ g rs ψrs (gkm g ij − gjm g ik ) + g ij Dkm + g ik Dmj + g im Djk ) = 0 ,
where
Dij = gjr g rs ψsi − gir g rs ψsj .
Next, permuting (10) cyclically in the indices m, j, k, we obtain
(11)
ejm + g im A
ekj + g ij A
emk
(n + 3)(Emijk + Ekimj + Ejikm ) + g ik A
−3L(g ij Dkm + g ik Dmj + g im Djk ) = 0 ,
which, by contraction with g ij , yields
emk = −3(n − 2)LDmk ,
(12)
(2n + 1)A
where
eij = Aij − Aji .
A
On the other hand, (10), together with (11), implies
(13)
nEmijk + g ik Ajm − g ij Akm
1
ejm + g im A
ekj + g ij A
emk )
−
(g A
n + 3 ik
n
L(g ik Dmj + g im Djk + g ij Dkm )
−
n+3
+ ηRimjk − nLRmijk
− L(n(gjm ψik − gkm ψij ) + g rs grs (g ik ψmj − g ij ψmk )
318
F. D E F E V E R AND M. D E S Z C Z
+ g rs ψrs (gkm g ij − gim g jk )) = 0 .
Contracting this with g ij and antisymmetrizing the resulting equality we
find
emk = −n(3n − 1) LDmk ,
(2n + 1)(n + 1)A
which, by (12), gives
emk = LDmk = 0 .
A
Now (13) turns into
Emijk =
1
1
(g Akm − g ik Ajm ) + ηRmijk + LRmijk
n ij
n
1
+ L (gjm ψik − gkm ψij ) + g rs grs (g ik ψmj − g ij ψmk )
n
1 rs
+ g ψrs (gkm g ij − gim g ik ) .
n
Finally, substituting this in (9), we obtain our assertion.
3. Main result
ψ
Proposition 3.1. Let f : (M, g)−→(M , g) be a geodesic mapping of
a pseudosymmetric manifold (M, g), dim M ≥ 3, onto a manifold (M , g).
Then the manifold (UR , g) is also pseudosymmetric.
P r o o f. Assume that the condition (1) holds on UR . Moreover, let
{x1 , . . . , xn } be a common coordinate system on UR and UR . Antisymmetrizing (8) in h, i and symmetrizing the resulting equality in the pairs h, i
and j, k we find
1
4(R.R)hijklm = − ηQ(g, R)hijklm
(14)
n
− 3Q(B, R)hijklm − 3Q(F, G)hijklm .
On the other hand, symmetrizing (8) in h, i we obtain
(15)
eil Rhmjk − B
eim Rhljk + B
ehl Rimjk − B
ehm Riljk
0=B
+ Fil Ghmjk − Fim Ghljk + Fhl Gimjk − Fhm Giljk ,
where
eil = Bil − 1 ηg il .
B
n
e
eij is a zero
We now prove that the tensor B with the local components B
e is non-zero at a point. Moreover, let V be a vectensor. Suppose that B
eij = ε, ε = ±1.
tor at this point with local components V i such that V i V j B
319
PSEUDOSYMMETRIC RIEMANNIAN MANIFOLDS
Transvecting (15) with V i and V l and antisymmetrizing the resulting equality in h, m we obtain
Rhmjk = −εV s V r Fsr Ghmjk ,
which easily gives Z(R) = 0, a contradiction.
Thus we see that (15) reduces to
Fil Ghmjk − Fim Ghijk + Fhl Gimjk − Fhm Giljk = 0 ,
which, by contraction with g hk and g mi , yields Fil = 0. Now (14) completes
the proof of our proposition.
ψ
Let f : (M, g)−→(M , g) be a geodesic mapping of a manifold (M, g)
onto a manifold (M , g). We note that if the tensor Z(R) vanishes at a point
x ∈ M then the tensor Z(R) also vanishes at the point f (x) ∈ M . This
remark, together with Proposition 3.1, implies the assertion of Theorem 1.1.
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INSTITUUT VOOR THEORETISCHE FYSICA
KATHOLIEKE UNIVERSITEIT LEUVEN
DEPARTMENT OF MATHEMATICS
AGRICULTURAL ACADEMY
CELESTIJNENLAAN 200D
C. NORWIDA 25
B-3030 LEUVEN, BELGIUM
50-375 WROCLAW, POLAND
Reçu par la Rédaction le 30.8.1990