C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. 2

VOL. LXII 1991 FASC. 2

A NOTE ON GEODESIC MAPPINGS

OF PSEUDOSYMMETRIC RIEMANNIAN MANIFOLDS

BY

FILIP D E F E V E R * (LEUVEN)

RYSZARD D E S Z C Z (WROC LAW)

1. Introduction. Let (M, g) be a connected n-dimensional , n ≥ 3, semi-Riemannian smooth manifold. We denote by ∇, e R, R, S and κ the Levi-Civit` a connection, the curvature tensor, the Riemann–Christoffel cur- vature tensor, the Ricci tensor and the scalar curvature of (M, g), respec- tively. We define on M the tensor fields R.R and Q(g, R) by the formulas

(R.R)(X 1 , X 2 , X 3 , X 4 ; X, Y )

= − R( e R(X, Y )X 1 , X 2 , X 3 , X 4 ) − R(X 1 , e R(X, Y )X 2 , X 3 , X 4 )

− R(X ₁ , X 2 , e R(X, Y )X 3 , X 4 ) − R(X 1 , X 2 , X 3 , e R(X, Y )X 4 ) , Q(g, R)(X 1 , X 2 , X 3 , X 4 ; X, Y )

= R((X ∧ Y )X 1 , X 2 , X 3 , X 4 ) + R(X 1 , (X ∧ Y )X 2 , X 3 , X 4 ) + R(X 1 , X 2 , (X ∧ Y )X 3 , X 4 ) + R(X 1 , X 2 , X 3 , (X ∧ Y )X 4 ) , where

(X ∧ Y )Z = g(Y, Z)X − g(X, Z)Y ,

X, Y, Z, X 1 , . . . , X 4 ∈ 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 U R = {x ∈ M | Z(R) 6= 0 at x}, where L is a function on U R and Z(R) = R − κ

n(n − 1) G

*Aspirant NFWO, Belgium.

with G defined by

G(X 1 , X 2 , X 3 , X 4 ) = g((X 1 ∧ X ₂ )X 3 , X 4 ) , X 1 , . . . , X 4 ∈ 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 essen- tially 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 man- ifolds 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 semi- Riemannian 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 mani- folds. 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 ; x ^j }. We denote by Γ _ij ^h , g ij , R ^h _ijk , R hijk and S ij the local components of the Levi- Civit` a connection ∇ and the local components of the tensors g, e R, R and S, respectively. Further, we denote by

(R.R) hijklm = ∇ m ∇ _l R hijk − ∇ _l ∇ _m R hijk

(2)

= −R rijk R ^r _hlm − R hrjk R ^r _ilm − R hirk R ^r _jlm − R hijr R ^r _klm , Q(g, R) hijklm = g hm R lijk + g im R hljk + g jm R hilk + g km R hijl

(3)

− g hl R mijk − g il R hmjk − g jl R himk − g kl R hijm , 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 )(X 1 , . . . , X k ; X, Y ) = T ((X ∧ A Y )X 1 , X 2 , . . . , X k )

+ T (X 1 , (X ∧ A Y )X 2 , . . . , X k ) + . . . + T (X 1 , . . . , X k−1 , (X ∧ A Y )X k ) , where X, Y, X 1 , . . . , X k ∈ 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 {x ¹ , . . . , x ⁿ }, Christoffel symbols and curvature tensors of (M, g) and (M , g) are related by

Γ ^h _ij = Γ _ij ^h + δ _i ^h ψ j + δ _j ^h ψ i , (4)

R ^h _ijk = R ^h _ijk + δ _j ^h ψ ik − δ ^h _k ψ ij , (5)

where

(6) ψ ij = ∇ j ψ i − ψ _i ψ j ,

(7) ψ i = 1

2(n + 1)

∂

∂x ⁱ

 log

det g det g    

 .

In the sequel such a geodesic mapping of (M, g) onto (M , g) will be 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 ^ψ called non-trivial on M if the covector field ψ with the local components ψ i

is non-zero.

R e m a r k. If f : (M, g) −→(M , g) is a geodesic mapping, then f (U ^ψ R ) = U _R . We can prove this using the fact that the Weyl projective curvature tensor W , defined by

W (X, Y )Z = e R(X, Y )Z − 1

n − 1 (S(Y, Z)X − S(X, Z)Y ) ,

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.

Lemma 2.1. Let f : (M, g) −→(M , g) be a geodesic mapping of a pseu- ^ψ dosymmetric manifold (M, g) onto a manifold (M , g) and let the condition R.R = LQ(g, R) be satisfied on U R . Then in a common coordinate system {x ¹ , . . . , x ⁿ } on U _R and U _R

(8) (R.R) hijklm = 1

n η(g _hl R mijk − g _hm R lijk )

+ B il R hmjk − B _im R hljk + B jl R himk − B _jm R hilk + B lk R hijm − B _km R hijl

+ F il G hmjk − F im G hljk − F jl G himk − F jm G hilk + F lk G hijm − F km G hijl , where

B ij = −Lg ij + ψ ij ,

F ij = − 1

n A ij − 1

n Lg ^rs (ψ rs g ij − g rs ψ ij ), A ij = ψ ir S ^r _j − ψ _rs R ^r _ij ^s ,

η = g ^rs (−g rs L + ψ rs ) .

P r o o f. Using the Ricci identity and (5) we obtain

∇ _m ∇ _l R ^s _ijk − ∇ _l ∇ _m R ^s _ijk = ∇ m ∇ _l R ^s _ijk − ∇ _l ∇ _m R ^s _ijk

+ ψ il R ^s _mjk − ψ _im R ^s _ljk + ψ jl R ^s _imk − ψ _jm R ^s _ilk + ψ kl R ^s _ijm − ψ _km R ^s _ijl + δ _k ^s (ψ ir R ^r _jlm + ψ jr R ^r _ilm ) − δ ^s _j (ψ ir R ^r _klm + ψ kr R ^r _ilm )

+ δ _l ^s ψ mr R ^r _ijk − δ ^s _m ψ lr R ^r _ijk ,

which, by making use of (5), (R.R) ^s _ijklm = LQ(g, R) ^s _ijklm and (3), turns into

∇ m ∇ l R ^h _ijk − ∇ l ∇ m R ^h _ijk = −L(δ _l ^h R mijk − δ _m ^h R lijk )

+ δ ^h _l E mijk − δ ^h _m E lijk + δ _k ^h (E ijlm + E jilm ) − δ _j ^h (E iklm + E kilm )

+ ψ il R ^h _mjk − ψ _im R ^h _ljk + ψ jl R ^h _imk − ψ _jm R ^h _ilk + ψ kl R ^h _ijm − ψ _km R ^h _ijl

− L(g _il (R ^h _mjk + δ ^h _k ψ mj − δ _j ^h ψ mk ) − g im (R ^h _ljk + δ _k ^h ψ lj − δ _j ^h ψ lk ) + g jl (R ^h _imk + δ _k ^h ψ im − δ _m ^h ψ ik ) − g jm (R ^h _ilk + δ _k ^h ψ il − δ _l ^h ψ ik ) + g kl (R ^h _ijm + δ _m ^h ψ ij − δ _j ^h ψ im ) − g km (R ^h _ijl + δ _l ^h ψ ij − δ ^h _j ψ il )) , where

E mijk = ψ mr R ^r _ijk . But this, by contraction with g _hs , gives

(9) (R.R) hijklm = −L(g _hl R mijk − g _hm R lijk )

+ g _hl E mijk − g _hm E lijk + g _hk (E ijlm + E jilm ) − g _hj (E iklm + E kilm ) + ψ il R hmjk − ψ _im R hljk + ψ jl R himk − ψ _jm R hilk + ψ kl R hijm − ψ _km R hijl

− L(g il (R hmjk + g _hk ψ mj − g _hj ψ mk ) − g im (R hljk + g _hk ψ lj − g _hj ψ lk ) + g jl (R himk + g _hk ψ im − g _hm ψ ik ) − g jm (R hilk + g _hk ψ il − g _hl ψ ik ) + g kl (R hijm + g _hm ψ ij − g _hj ψ im ) − g km (R hijl + g _hl ψ ij − g _hj ψ il )) . Symmetrizing (9) in h, i we obtain

g _hl E mijk − g _hm E lijk + g _il E mhjk − g _im E lhjk

+ g _hk (E ijlm + E jilm ) + g _ik (E hjlm + E jhlm )

− g _hj (E iklm + E kilm ) − g _ij (E hklm + E khlm )

+ ψ il R hmjk − ψ _im R hljk + ψ hl R imjk − ψ _hm R iljk

− L(g _il R mhjk + g _hl R mijk − g _hm R lijk − g _im R lhjk )

− L(g _il (R hmjk + g _hk ψ mj − g _hj ψ mk ) + g hl (R imjk + g _ik ψ mj − g _ij ψ mk )

− g im (R hljk + g _hk ψ lj − g _hj ψ lk ) − g hm (R iljk + g _ik ψ lj − g _ij ψ lk ) + g jl (g _hk ψ im − g _im ψ hk + g _ik ψ hm − g _hm ψ ik )

− g _jm (g _hk ψ il − g _il ψ hk + g _ik ψ hl − g _hl ψ ik ) + g kl (g _hm ψ ij − g _ij ψ hm + g _im ψ hj − g _hj ψ im )

− g _km (g _hl ψ ij − g _ij ψ hl + g _il ψ hj − g _hj ψ il )) = 0 . Contracting this with g ^hl and using the identity

(R.g) imjk = − g is R ^s _mjk − g ms R ^s _ijk = −g is R ^s _mjk − g ms R ^s _ijk

− g _is (δ ^s _j ψ mk − δ ^s _k ψ mj ) − g ms (δ ^s _j ψ ik − δ ^s _k ψ ij ) we get

(n + 1)E mijk + E jikm + E kimj

(10)

+ g _ik A jm − g _ij A km + ηR imjk − nLR _mijk

− L(n(g _jm ψ ik − g _km ψ ij ) + g ^rs g rs (g _ik ψ mj − g _ij ψ mk )

+ g ^rs ψ rs (g km g _ij − g _jm g _ik ) + g _ij D km + g _ik D mj + g _im D jk ) = 0 , where

D ij = g jr g ^rs ψ si − g ir g ^rs ψ sj .

Next, permuting (10) cyclically in the indices m, j, k, we obtain (11) (n + 3)(E mijk + E kimj + E jikm ) + g _ik A e jm + g _im A e kj + g _ij A e mk

−3L(g _ij D km + g _ik D mj + g _im D jk ) = 0 , which, by contraction with g ^ij , yields

(12) (2n + 1) e A mk = −3(n − 2)LD mk , where

A e ij = A ij − A _ji . On the other hand, (10), together with (11), implies

nE mijk + g _ik A jm − g _ij A km

(13)

− 1

n + 3 (g _ik A e jm + g _im A e kj + g _ij A e mk )

− n

n + 3 L(g _ik D mj + g _im D jk + g _ij D km ) + ηR imjk − nLR mijk

− L(n(g _jm ψ ik − g _km ψ ij ) + g ^rs g rs (g _ik ψ mj − g _ij ψ mk )

+ g ^rs ψ rs (g km g _ij − g _im g _jk )) = 0 .

Contracting this with g ^ij and antisymmetrizing the resulting equality we find

(2n + 1)(n + 1) e A mk = −n(3n − 1) LD mk , which, by (12), gives

A e mk = LD mk = 0 . Now (13) turns into

E mijk = 1

n (g _ij A km − g _ik A jm ) + 1

n ηR mijk + LR mijk

+ L



(g jm ψ ik − g _km ψ ij ) + 1

n g ^rs g rs (g _ik ψ mj − g _ij ψ mk ) + 1

n g ^rs ψ rs (g km g _ij − g _im g _ik )

 . Finally, substituting this in (9), we obtain our assertion.

3. Main result

Proposition 3.1. Let f : (M, g) −→(M , g) be a geodesic mapping of ^ψ a pseudosymmetric manifold (M, g), dim M ≥ 3, onto a manifold (M , g).

Then the manifold (U _R , g) is also pseudosymmetric.

P r o o f. Assume that the condition (1) holds on U R . Moreover, let {x ¹ , . . . , x ⁿ } be a common coordinate system on U _R and U _R . Antisym- metrizing (8) in h, i and symmetrizing the resulting equality in the pairs h, i and j, k we find

4(R.R) hijklm = − 1

n ηQ(g, R) hijklm

(14)

− 3Q(B, R) _hijklm − 3Q(F, G) _hijklm . On the other hand, symmetrizing (8) in h, i we obtain

0 = e B il R hmjk − e B im R hljk + e B hl R imjk − e B hm R iljk

(15)

+ F il G hmjk − F _im G hljk + F hl G imjk − F _hm G iljk , where

B e il = B il − 1 n ηg _il .

We now prove that the tensor e B with the local components e B ij is a zero

tensor. Suppose that e B is non-zero at a point. Moreover, let V be a vec-

tor at this point with local components V ⁱ such that V ⁱ V ^j B e ij = ε, ε = ±1.

Transvecting (15) with V ⁱ and V ^l and antisymmetrizing the resulting equal- ity in h, m we obtain

R hmjk = −εV ^s V ^r F sr G hmjk , which easily gives Z(R) = 0, a contradiction.

Thus we see that (15) reduces to

F il G hmjk − F _im G hijk + F hl G imjk − F _hm G iljk = 0 ,

which, by contraction with g ^hk and g ^mi , yields F il = 0. Now (14) completes the proof of our proposition.

Let f : (M, g) −→(M , g) be a geodesic mapping of a manifold (M, g) ^ψ 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 DEPARTMENT OF MATHEMATICS KATHOLIEKE UNIVERSITEIT LEUVEN AGRICULTURAL ACADEMY

CELESTIJNENLAAN 200D C. NORWIDA 25

B-3030 LEUVEN, BELGIUM 50-375 WROC LAW, POLAND

Re¸ cu par la R´ edaction le 30.8.1990