# The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups

@article{Buttsworth2016ThePR, title={The prescribed Ricci curvature problem on three‐dimensional unimodular Lie groups}, author={Timothy Buttsworth}, journal={Mathematische Nachrichten}, year={2016}, volume={292}, pages={747 - 759} }

Let G be a three‐dimensional unimodular Lie group, and let T be a left‐invariant symmetric (0,2)‐tensor field on G. We provide the necessary and sufficient conditions on T for the existence of a pair (g,c) consisting of a left‐invariant Riemannian metric g and a positive constant c such that Ric(g)=cT , where Ric(g) is the Ricci curvature of g. We also discuss the uniqueness of such pairs and show that, in most cases, there exists at most one positive constant c such that Ric(g)=cT is solvable…

## 12 Citations

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## References

SHOWING 1-10 OF 21 REFERENCES

### Left invariant metrics and curvatures on simply connected three‐dimensional Lie groups

- Mathematics
- 2009

For each simply connected three‐dimensional Lie group we determine the automorphism group, classify the left invariant Riemannian metrics up to automorphism, and study the extent to which curvature…

### Metrics with Prescribed Ricci Curvature of Constant Rank

- Mathematics
- 1999

In this paper, we give local existence results in the real-analytic category for the non-linear partial differential equation arising from the problem of finding Riemannian metrics with prescribed…

### Ricci curvature in the neighborhood of rank-one symmetric spaces

- Mathematics
- 2001

We study the Ricci curvature of a Riemannian metric as a differential operator acting on the space of metrics close (in a weighted functional spaces topology) to the standard metric of a rank-one…

### On solutions of the Ricci curvature equation and the Einstein equation

- Mathematics
- 2009

We consider the pseudo-Euclidean space (Rn, g), with n ≥ 3 and gij = δij εi, εi = ±1, where at least one εi = 1 and nondiagonal tensors of the form T = Σijfijdxidxj such that, for i ≠ j, fij (xi, xj)…

### Ricci iteration on homogeneous spaces

- MathematicsTransactions of the American Mathematical Society
- 2019

The Ricci iteration is a discrete analogue of the Ricci flow. We give the first study of the Ricci iteration on a class of Riemannian manifolds that are not Kähler. The Ricci iteration in the…

### Studies of Some Curvature Operators in a Neighborhood of an Asymptotically Hyperbolic Einstein Manifold

- Mathematics
- 2002

Abstract On an asymptotically hyperbolic Einstein manifold ( M , g 0 ) for which the Yamabe invariant of the conformal structure on the boundary at infinity is nonnegative, we show that the operators…

### The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional lie groups. The unimodular case

- Mathematics
- 2009

We obtain a complete classification of all the possible values of the signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional unimodular Lie groups.

### On Ricci eigenvalues of locally homogeneous Riemannian 3-manifolds

- Mathematics
- 1996

We find the necessary and sufficient conditions for three constants ϱ1, ϱ2, ϱ3 ∈ ℝ3 to be the principal Ricci curvatures of some 3-dimensional locally homogeneous Riemannian space.