Walker Osserman Metric of Signature (3, 3)

Abstract

A Walker m-manifold is a pseudo-Riemannian manifold, which admits a field of parallel null r-planes, with $r{\leqslant } \frac {m}{2}$. The Riemann extension is an important method to produce Walker metric on the cotangent bundle T ^∗ M of any affine manifold (M, ∇). In this paper, we investigate the torsion-free affine manifold (M, ∇) and their Riemann extension $(T^* M,\bar {g})$ as concerns heredity of the Osserman condition.

Dedicated to Prof. Mahouton Norbert Hounkonnou on the occasion of his 60th Birthday

Appendices

Appendix 1: Components of the Curvature Tensor

The non-zero components of the curvature tensor of the affine connection (3.1) are given by

$$\displaystyle \begin{aligned} R(\partial_1,\partial_2)\partial_1 &= (\partial_1 f_2 -\partial_2 f_1 +f^{2}_{2} -f_1 f_4)\partial_2 \\ R(\partial_1,\partial_2)\partial_2 &= (\partial_1 f_4 -\partial_2 f_2)\partial_2 \\ R(\partial_1,\partial_2)\partial_3 &= (\partial_1 f_5 -\partial_2 f_3 +f_2f_5 -f_3f_4)\partial_2 \\ R(\partial_1,\partial_3)\partial_1 &= (\partial_1 f_3 -\partial_3 f_1 +f_2 f_3 -f_1f_5)\partial_2\\ R(\partial_1,\partial_3)\partial_2 &= (\partial_1 f_5 -\partial_3 f_2)\partial_2 \\ R(\partial_1,\partial_3)\partial_3 &= (\partial_1 f_6 -\partial_3 f_3 +f_2f_6 -f_3 f_5)\partial_2\\ R(\partial_2,\partial_3)\partial_1 &= (\partial_2 f_3 -\partial_3 f_2 +f_3f_4 -f_2f_5)\partial_2\\ R(\partial_2,\partial_3)\partial_2 &= (\partial_2 f_5 -\partial_3 f_4)\partial_2\\ R(\partial_2,\partial_3)\partial_3 &= (\partial_2 f_6 -\partial_3 f_5 +f_4f_6 -f^{2}_{5})\partial_2. \end{aligned} $$

Appendix 2: Osserman Geometry

Let R be the curvature operator of a Riemannian manifold (M, g) of dimension m. The Jacobi operator $\mathcal {J}(x):y\mapsto R(y,x)x$ is the self-adjoint endomorphism of the tangent bundle. Following the seminal work of Osserman [20], one says that (M, g) is Osserman if the eigenvalues of $\mathcal {J}$ are constant on the unit sphere bundle

$$\displaystyle \begin{aligned} \begin{array}{rcl} S(M,g):=\{X\in TM: g(X,X)=1\}. \end{array} \end{aligned} $$

Work of Chi [5], of Gilkey et al. [15], and of Nikolayevsky [16, 17] show that any complete and simply connected Osserman manifold of dimension m ≠ 16 is a rank-one symmetric space; the 16-dimensional setting is exceptional and the situation is still not clear in that setting although there are some partial result due, again, to Nikolayevsky [18].

Suppose (M, g) is a pseudo-Riemannian manifold of signature (p, g) for p > 0 and q > 0. The pseudo-sphere bundles are defined by setting

$$\displaystyle \begin{aligned} \begin{array}{rcl} S^{\pm}(M,g):=\{X\in TM: g(X,X)= \pm 1\}. \end{array} \end{aligned} $$

One says that (M, g) is spacelike (resp. timelike) Osserman if the eigenvalues of $\mathcal {J}$ are constant on S ⁺(M, g) (resp. S ⁻(M, g)). The situation is rather different here as the Jacobi operator is no longer diagonalizable and can have nontrivial Jordan normal form as shown by Garcá-Ró et al. [13]. We refer to [14] for more information on Osserman manifolds.