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**Volume-preserving geodesic symmetries on four-dimensional Kähler manifolds.**
*(English)*
Zbl 0605.53031

Differential geometry, Proc. 2nd Int. Symp., Peñiscola/Spain 1985, Lect. Notes Math. 1209, 275-291 (1986).

[For the entire collection see Zbl 0594.00011.]

In an earlier paper, the reviewer has proved that all three-dimensional Riemannian manifolds with volume-preserving local geodesic symmetries are locally homogeneous. Hence the full (local) list of these spaces has been derived, including three classes of non-symmetric spaces.

The four-dimensional case is much more difficult to investigate. The present authors solve a special case completely in the following theorem: let (M,g,J) be a connected four-dimensional Kähler-manifold such that all local geodesic symmetries are volume-preserving. Then (M,g) is locally symmetric.

The proof is far from being easy; yet, it still depends heavily on the following (unpublished) result by A. Derdziński: Let (M,g) be a four- dimensional Einstein manifold such that the Weyl tensor \(W\in C^{\infty}(End \wedge^ 2M)\) has constant eigenvalues. Then (M,g) is locally symmetric.

In an earlier paper, the reviewer has proved that all three-dimensional Riemannian manifolds with volume-preserving local geodesic symmetries are locally homogeneous. Hence the full (local) list of these spaces has been derived, including three classes of non-symmetric spaces.

The four-dimensional case is much more difficult to investigate. The present authors solve a special case completely in the following theorem: let (M,g,J) be a connected four-dimensional Kähler-manifold such that all local geodesic symmetries are volume-preserving. Then (M,g) is locally symmetric.

The proof is far from being easy; yet, it still depends heavily on the following (unpublished) result by A. Derdziński: Let (M,g) be a four- dimensional Einstein manifold such that the Weyl tensor \(W\in C^{\infty}(End \wedge^ 2M)\) has constant eigenvalues. Then (M,g) is locally symmetric.

Reviewer: O.Kowalski

### MSC:

53C55 | Global differential geometry of Hermitian and Kählerian manifolds |

53C20 | Global Riemannian geometry, including pinching |