Higher order Osserman pseudo-Riemannian manifolds of neutral signature (2, 2)

Higher order Osserman pseudo-Riemannian manifolds of neutral signature (2, 2)

C˘at˘alin S¸terbet¸i

Abstract

In this paper we construct a family of pseudo-Riemannian metrics of neutral signature (2,2) which leads to k-Osserman manifolds for all k admissible. For these manifolds the generalized Jacobi operator is 2-nilpotent. Conditions for locally symmetry on the considered manifolds are established.

Mathematics Subject Classification:53A45, 53C35, 53C50.

Key words:generalized Jacobi operator, locally symmetric.

Let (M, g) be a pseudo-Riemannian manifold of signature (p, q) and dimension n = p+q. Let R(·,·) be the Riemannian curvature operator. The Jacobi operator J(X) :Y →R(Y, X)X is a self-adjoint operator and it plays an important role in the study of geodesic variations.

LetS^±(M) be the pseudo-sphere bundles of unit spacelike (+) and timelike (−) vectors for the manifold (M, g). Then (M, g) is said to bespacelike Osserman (re- spectively timelike Osserman) if the eigenvalues of J(·) are constant on S⁺(M, g) (respectively onS⁻(M, g)). The notions spacelike Osserman and timelike Osserman are equivalent and if (M, g) is either of them, then (M, g) is said to be Osserman.

In this paper we study the higher order Jacobi operator, which was first defined by Stanilov and Videv ([9]) in the Riemannian setting. This definition was extended to semi-Riemannian geometry in [6]. Letπbe a nondegeneratek-plane inTpM, with orthonormal basis {e1, . . . , ek}, where (M, g) is a pseudo-Riemannian manifold of signature (p, q). The generalized Jacobi operator is defined by

JR(π) = Xk i=1

g(ei, ei)R(·, ei)ei.

We say that a pair of integers (r, s) is an admissible pair for TpM if 0 ≤ r ≤ p, 0≤s≤qand 1≤s+r≤p+q−1. This means that the GrassmannianGr(r,s)(TpM) of all non-degenerate planes in TpM of signature (r, s) is non-empty and does not consist of a single point.

Let (r, s) be an admissible pair. We say that (M, g) is Ossermann of type (r, s) in p∈M if the eigenvalues of the operatorJR(π) do not depend on the choice of plane π∈Gr_(r,s)(TpM).

Balkan Journal of Geometry and Its Applications, Vol.10, No.1, 2005, pp. 175-178.∗

c

°Balkan Society of Geometers, Geometry Balkan Press 2005.

176 C. Sterbet¸i P. Gilkey shows that if (M, g) is Osserman of type (r, s) then it is Osserman of type (˜r,s) for all admissible pairs (˜˜ r,˜s) satisfyingr+s= ˜r+ ˜s([3], [4]). Thus, only the dimensionk=r+sof planesπis relevant and we simply talk aboutk-Osserman. A semi-Riemannian manifold (M, g) is said to be ak-Osserman manifold if for all points p∈M, (M, g) isk-Osserman inpwith the eigenvalue structure ofJRp(·) independent of the chosen pointp.

Let M =R⁴ with coordinates (x, y) = (x¹, x², y¹, y²). Then X =Span{∂₁^x, ∂₂^x} andY =Span{∂₁^y, ∂₂^y} define two distributions ofT M. The splittingT M =XL

Y is just the usual splittingTR⁴=TR²L

TR². We define a semi-Riemannian metric of neutral signature (2,2) by setting

g(f1,f2,h) = y¹f1(x¹)dx¹⊗dx¹+y²f2(x²)dx²⊗dx²+ + h(x¹, x²)[dx¹⊗dx²+dx²⊗dx¹]+

+ a[dx¹⊗dy¹+dy¹⊗dx¹+dx²⊗dy²+dy²⊗dx²], (0.1)

where a ∈ R^∗ and f1, f2, h are smooth real valued functions. The coefficients of g(f1,f2,h)depend onxandy. Furthermore, the distributionYis totally isotropic with respect tog(f1,f2,h).

Lemma 1 The only nonvanishing covariant derivatives are given by 5∂₁^x∂₁^x = −_2a¹f1(x¹)∂₁^x+

h1

2ay¹f₁⁰(x¹) +_2a^y12f₁²(x¹) i

∂^y₁+ + £₁

a ∂h

∂x¹(x¹, x²) +_2a¹2f1(x¹)h(x¹, x²)¤

∂₂^y, 5∂₂^x∂₂^x = −_2a¹f2(x²)∂₂^x+£ ₁

2a²f2(x²)h(x¹, x²) +_a¹_∂x^∂h2(x¹, x²)¤

∂₁^y+

+ h

1

2ay²f₂⁰(x²) +_2a^y22f₂²(x²)i

∂^y₂, 5∂₁^x∂₁^y = _2a¹f1∂₁^y,

5∂₂^x∂₂^y = _2a¹f2∂₂^y. (0.2)

From (0.1) we have the following:

Proposition 1 The only nonvanishing components of the curvature tensor of(R⁴, g_(f1,f2,h)) are given by

R(∂₁^x, ∂₂^x)∂₁^x = −¹_a h ∂²h

∂x¹∂x² +_2a¹f2∂h

∂x¹ +_2a¹f1 ∂h

∂x² +_4a¹2f1f2h i

∂^y₂, R(∂₁^x, ∂₂^x)∂₂^x = ¹_a

h ∂²h

∂x¹∂x² +_2a¹f2 ∂h

∂x¹ +_2a¹f1∂h

∂x² +_4a¹2f1f2h i

∂₂^y. (0.3)

Theorem 1 Let p≥2. Then(M, g(f1,f2,h))isk-Osserman for every admissiblek.

Proof. Let be X1, X2, X3 coordinate vector fields. By proposition 1, J(X1)X3 = R(X3, X1)X1= 0 if X1∈ Y. ThusY ⊂Ker(J(X1)). Furthermore,range(J(X2))⊂ span{R(∂_i^x, ∂_j^x)∂_k^x} ⊂ Y. ThusJ(X1)J(X2) = 0.

If {X₁, X₂, . . . , X_k} is an orthonormal basis for π ∈ Gr_(r,s)(M, g_(f1,f2,h)), then we have

J(π)²= Xk

i,j=1

g_(f1,f2,h)(Xi, Xi)g_(f1,f2,h)(Xj, Xj)J(Xi)J(Xj) = 0.

Higher order Osserman pseudo-Riemannian manifolds ... 177 Theorem 2 Let p≥2. The manifold (R⁴, g(f1,f2,h))is a locally symmetric space if and only if the functions f1,f2,h are solutions of the following partial differential equations inR²:

∂Φ

∂x^k +fk

2aΦ = 0, k= 1,2, (0.4)

where we note

Φ(x¹, x²) = 1 a

· ∂²h

∂x¹∂x² + 1 2af2∂h

∂x¹ + 1 2af1 ∂h

∂x² + 1 4a²f1f2h

¸ .

Proof.If we take in account this notation, we obtain by (0.3) R(∂₁^x, ∂₂^x)∂_k^x= (−1)^kΦ(x¹, x²)∂_3−k^y , k= 1,2.

LetXk=α^k_i∂_i^x, k= 1,4,i= 1,4. The condition∇X1R(X2, X3)X4= 0 leads to

∇_α¹

i∂^x_iR(α²_j∂_j^x, α³_l∂^x_l)α⁴_s∂_s^x= 0, i, j, k, s= 1,4. Equivalently,

α¹₂α²₁α³₂α⁴₁∇∂^x₂R(∂₁^x, ∂₂^x)∂₁^x+α¹₁α²₁α³₂α⁴₂∇∂^x₁R(∂₁^x, ∂₂^x)∂₂^x+ +α¹₂α²₂α³₁α⁴₁∇∂₂^xR(∂₂^x, ∂₁^x)∂₁^x+α¹₁α²₂α³₁α⁴₂∇∂₁^xR(∂₂^x, ∂₁^x)∂₂^x= 0. But

∇∂₁^xR(∂₁^x, ∂₂^x)∂₂^x=−∇∂₁^xR(∂₂^x, ∂₁^x)∂₂^x=∇∂₁^xΦ∂₁^y= µ∂Φ

∂x¹ +f1

2aΦ

¶

∂₁^y

∇∂₂^xR(∂^x₁, ∂^x₂)∂^x₁ =−∇∂₂^xR(∂^x₂, ∂^x₁)∂^x₁ =− µ∂Φ

∂x² +f2

2aΦ

¶

∂₂^y.

The proof is complete.

Corollary 1 If h(x¹, x²) ≡ C (h is a constant function), the conditions (0.4) for locally symmetry becames

½ f₁⁰(x¹)f2(x²) +_2a¹f₁²(x¹)f2(x²) = 0, f₂⁰(x²)f1(x¹) +_2a¹f₂²(x²)f1(x¹) = 0. (0.5)

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C˘at˘alin S¸terbet¸i

University of Craiova,Department of Applied Mathematics Craiova, 1100, Romˆania

e-mail address: [email protected]