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We classify homogeneous four-dimensional manifolds satisfying a curvature condition which naturally generalizes the Einstein one. Our work leads to new examples of self-similar solutions of the two-loop renormalization group flow Supported by projects PID2019-105138GB-C21(AEI/FEDER, Spain), and ED431C 2019/10, ED431F 2020/04 (Xunta de Galicia, Spain) SI

We complete the classification of locally conformally flat Kahler and para-Kahler manifolds, describing all possible non-flat curvature models for Kahler and para-Kahler surfaces.

We show that four-dimensional homogeneous Kähler Ricci solitons are Einstein, rigid or an algebraic Ricci soliton on the only 3 3 -symmetric space.

We show the existence of non-Einstein homogeneous critical metrics for any quadratic curvature functional in dimension three.

We give a new short self-contained proof of the result of Opozda (Differ Geom Appl 21:173–198, 2004) classifying the locally homogeneous torsion free affine surfaces and the extension to the case of surfaces with torsion due to Arias-Marco and Kowalski (Monatsh Math 153:1–18, 2008). Our approach rests on a direct analysis of the affine Killing equations and is quite different than the approaches taken previously in the literature.

Homogeneous conformally Einstein manifolds are classified in dimension four. As a consequence we show that any homogeneous strictly Bach-flat four-dimensional manifold is homothetic to one of the examples constructed by Abbena, Garbiero and Salamon in [1] .

We classify four-dimensional homogeneous manifolds satisfying some generalized Einstein conditions.

It is shown that locally conformally flat weakly-Einstein manifolds are either locally symmetric, and hence a product $$N_1^m(c)\times N_2^m(-c)$$ , or otherwise they are locally homothetic to some specific warped product metrics $${\mathcal {I}}\times _fN(c)$$ . As an application we classify weakly-Einstein hypersurfaces in the Euclidean space.

Three-dimensional Brinkmann waves which are critical for quadratic curvature functionals are determined. Generically, if the metric is critical for some functional then it is critical for all of them. In contrast, there are four special functionals that do not share critical metrics with any other quadratic functional. It is also shown that these metrics provide explicit solutions for different massive gravity models.

We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete classification in dimension three., Comment: 19 pages. Updated version

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.

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.

We examine the topology of various spaces of locally homogeneous affine manifolds which arise from the classification result of Opozda [B. Opozda, A classification of locally homogeneous connections on 2-dimensional manifolds, Differential Geom. Appl. 21 (2004), 173-198.] as orbits of the action of $GL(2,\mathbb{R})$ (Type $\mathcal{A}$) and the $ax+b$ group (Type $\mathcal{B}$). We determine the topology of the spaces of Type $\mathcal{A}$ models in relation to the rank of the Ricci tensor. We determine the topology of the spaces of Type $\mathcal{B}$ models which either are flat or where the Ricci tensor is alternating.

Motivated by the construction of Bach flat neutral signature Riemannian extensions, we study the space of parallel trace-free tensors of type (1, 1) on an affine surface. It is shown that the existence of such a parallel tensor field is characterized by the recurrence of the symmetric part of the Ricci tensor.

We study the affine quasi-Einstein Equation for homogeneous surfaces. This gives rise through the modified Riemannian extension to new half conformally flat generalized quasi-Einstein neutral signature $(2,2)$ manifolds, to conformally Einstein manifolds and also to new Einstein manifolds through a warped product construction.

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.

k-Curvature homogeneous three-dimensional Walker metrics are described for . This allows a complete description of locally homogeneous three-dimensional Walker metrics, showing that there exist exactly three isometry classes of such manifolds. As an application one obtains a complete description of all locally homogeneous Lorentzian manifolds with recurrent curvature. Moreover, potential functions are constructed in all the locally homogeneous manifolds resulting in steady gradient Ricci and Cotton solitons.

Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kahler–Weyl structure; this structure is locally conformally Kahler if and only if the alternating Ricci tensor ρa vanishes. The tensor ρa takes values in a certain representation space. In this paper, we show that any algebraic possibility Ξ in this representation space can in fact be geometrically realized by a left-invariant Kahler–Weyl structure on a 4-dimensional Lie group in either the Hermitian or the para-Hermitian setting.

The non-existence of non-trivial conformally symmetric manifolds in the three-dimensional Riemannian setting is shown. In Lorentzian signature, a complete local classification is obtained. Furthermore, the isometry classes are examined.

We construct examples of Bach-flat gradient Ricci solitons which are neither half conformally flat nor conformally Einstein.

It is shown that indefinite strictly almost Kahler and opposite Kahler structures (J, J′) on a four-dimensional manifold with J-invariant Ricci operator are rigid, thus extending a previous result of Apostolov, Armstrong and Draghici from the positive definite case to the indefinite one. In contrast to this, examples of nonhomogeneous four-dimensional manifolds which admit strictly almost paraKahler and opposite paraKahler structures \({(\mathfrak{J},\mathfrak{J}^{\prime})}\) with \({\mathfrak{J}}\) -invariant Ricci operator are shown.

We exhibit Osserman metrics with non-nilpotent Jacobi operators and with non-trivial Jordan normal form in neutral signature (n,n) for any n which is at least 3. These examples admit a natural almost para-Hermitian structure and are semi para-complex Osserman with non-trivial Jordan normal form as well; they neither satisfy the third Gray identity nor are they integrable.

We use the modified Riemannian extension of an affine surface to construct Bach flat manifolds. As all these examples have vanishing scalar invariants, we shall construct scalar invariants which are not of Weyl type to distinguish them. We illustrate these phenomena in the context of homogeneous affine surfaces.We use the modified Riemannian extension of an affine surface to construct Bach flat manifolds. As all these examples have vanishing scalar invariants, we shall construct scalar invariants which are not of Weyl type to distinguish them. We illustrate these phenomena in the context of homogeneous affine surfaces.