We study the affine quasi-Einstein equation, a second order linear homogeneous equation, which is invariantly defined on any affine manifold. We prove that the space of solutions is finite-dimensional, and its dimension is a strongly projective invariant. Moreover the maximal dimension is shown to be achieved if and only if the manifold is strongly projectively flat.2010 Mathematics Subject Classification. 53C21, 53B30, 53C24, 53C44.
The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighbourhood of any point where the gradient of the potential function is non-null. In opposition, if the gradient of the potential function is null, then the soliton is a steady traceless κ-Einstein soliton and is realized on the cotangent bundle of an affine surface.
We provide classification results for and examples of half conformally flat generalized quasi Einstein manifolds of signature (2, 2). This analysis leads to a natural equation in affine geometry called the affine quasi-Einstein equation that we explore in further detail.2010 Mathematics Subject Classification. 53C21, 53B30, 53C24, 53C44.
We investigate the local structure of four-dimensional Lorentzian quasi-Einstein manifolds under conditions on the Weyl tensor. We show that if the Weyl tensor is harmonic and the potential function preserves this harmonicity then, in the isotropic case, the manifold is necessarily a pp-wave. Using the quasi-Einstein equation, further conclusions are obtained for pp-waves. In particular, we show that a four-dimensional pp-wave is conformally Einstein if and only if it is locally conformally flat or has harmonic Weyl tensor.
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