Homothety Curvature Homogeneity

We examine the difference between several notions of curvature homogeneity and show that the notions introduced by Kowalski and Van\v{z}urov\'a are genuine generalizations of the ordinary notion of $k$-curvature homogeneity. The homothety group plays an essential role in the analysis

Homogeneous affine surfaces: affine Killing vector fields and Gradient Ricci solitons

The homogeneous affine surfaces have been classified by Opozda. They may be grouped into 3 families, which are not disjoint. The connections which arise as the Levi-Civita connection of a surface with a metric of constant Gauss curvature form one family; there are, however, two other families. For a surface in one of these other two families, we examine the Lie algebra of affine Killing vector fields and we give a complete classification of the homogeneous affine gradient Ricci solitons. The rank of the Ricci tensor plays a central role in our analysis

Half conformally flat gradient Ricci almost solitons

The local structure of half conformally flat gradient Ricci almost solitons is investigated, showing that they are locally conformally flat in a neighborhood 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 $\kappa$-Einstein soliton and is realized on the cotangent bundle of an affine surface

Locally conformally flat Lorentzian quasi-Einstein manifolds

We show that locally conformally flat quasi-Einstein manifolds are globally conformally equivalent to a space form or locally isometric to a $pp$-wave or a warped product

Conformal geometry of non-reductive four-dimensional homogeneous spaces

We classify non-reductive four-dimensional homogeneous conformally Einstein manifolds.Comment: New version correcting some inaccuracies in the original pape

Homogeneous Ricci almost solitons

It is shown that a locally homogeneous proper Ricci almost soliton is either of constant sectional curvature or a product $\mathbb{R}\times N(c)$, where $N(c)$ is a space of constant curvature

Geometric properties of generalized symmetric spaces

It is shown that four-dimensional generalized symmetric spaces can be naturally equipped with some additional structures defined by means of their curvature operators. As an application, those structures are used to characterize generalized symmetric spaces

The structure of the Ricci tensor on locally homogeneous Lorentzian gradient Ricci solitons

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 versio

Solutions to the affine quasi-Einstein equation for homogeneous surfaces

We examine the space of solutions to the affine quasi--Einstein equation in the context of homogeneous surfaces. As these spaces can be used to create gradient Yamabe solitions, conformally Einstein metrics, and warped product Einstein manifolds using the modified Riemannian extension, we provide very explicit descriptions of these solution spaces. We use the dimension of the space of affine Killing vector fields to structure our discussion as this provides a convenient organizational framework.Comment: 5 figure

Affine surfaces which are K\"ahler, para-K\"ahler, or nilpotent K\"ahler

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