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We examine the harmonicity of some natural maps associated to the tangent and cotangent bundles, providing some new examples of proper biharmonic maps between pseudo-Riemannian manifolds.

A local description of proper Hermitian–Walker structures that are locally conformally Kahler, self-dual, ∗-Einstein or Einstein is given. This is used to supply examples of indefinite Einstein strictly almost Hermitian structures showing that an integrability result in [K.-D. Kirchberg, Integrability conditions for almost Hermitian and almost Kahler 4-manifolds, arXiv:math.DG/0605611 ] does not hold for metrics of signature ( 2 , 2 ) .

It is shown that any proper almost Hermitian structure on a Walker 4-manifold is isotropic Kahler. Moreover, a local description of proper almost Kahler structures that are self-dual, ∗ -Einstein or Einstein is given and it is proved that any proper strictly almost Kahler Einstein structure is self-dual, Ricci flat and ∗ -Ricci flat. This is used to supply examples of flat indefinite non-Kahler almost Kahler structures.

A local characterization of Schwarzschild metrics is made by showing the space-time is locally a 2 by 2 warped product and admitting a static reference frame on its certain open subsets under some assumptions on the global analytic structure and stress-energy tensor of the space-time, such as, assuming the existence of solutions to certain partial differential equations and the existence of a radiation stress-energy tensor consistent with these solutions on the space-time.

A rigid singularity theorem for spacetimes admitting irrotational reference frames is proven as an extension of a result (Theorem 3.5) of Petersen and Walschap [Observer fields and the strong energy condition, Class. Quantum Grav. 13 (1996) 1901–1908.].

Null and infinitesimal isotropy are defined for semi-Riemannian manifolds in a more general context. A theorem of Karcher is extended to Semi-Riemannian manifolds in a more general setting. Also, by using this theorem, a characterization of static blackhole metrics can be made as well as a characterization of Robertson-Walker metrics as Karcher made.