594 research outputs found
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 -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 -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 -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 , where
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
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
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