On a Generalization of Curvature Homogeneous Spaces

$g$-natural metrics of constant curvature on unit tangent sphere bundles

“…A locally homogeneous space is curvature homogeneous, but there are many well-known examples of curvature homogeneous Riemannian manifolds which are not locally homogeneous. We may refer to [9] for further results and references concerning curvature homogeneous manifolds, especially in dimension three. If dimM = 3, then curvature homogeneity is equivalent to the constancy of the Ricci eigenvalues.…”

Homogeneity of Lorentzian three-manifolds with recurrent curvature

“…It was shown in [23] that the existence of a linear homothety Φ p,q : T p M → T q M (i.e., Φ * p,q g q = λ 2 p,q g p ) such that Φ * p,q ∇ l R q = ∇ l R p is equivalent to the existence of a linear isometry ϕ p,q : T p M → T q M and 0 = λ p,q ∈ R so λ l+2 p,q ϕ * p,q ∇ l R q = ∇ l R p . Hence, (M, g) is Kowalski-Vanžurová k-curvature homogeneous if for any two points there exists a linear isometry ϕ p,q between the corresponding tangent spaces such that λ p,q l+2 ϕ * p,q ∇ l R q = ∇ l R p , for all l = 0, .…”

“…As any homogeneous Lorentzian three‐manifold with nilpotent Ricci operator is a Walker manifold , it follows from Theorem that any non‐symmetric locally conformally flat homogeneous three‐manifold with nilpotent Ricci operator is locally isometric to the manifold . Further observe that all non‐conformally flat left‐invariant metrics on three‐dimensional Lie groups (see, for example the discussion in , ) with nilpotent Ricci operators are locally isometric to the manifold . Remark There is a different notion of curvature homogeneity that is due to Kowalski and Vanžurová , . Motivated by their seminal work, we say that a manifold is Kowalski‐Vanžurová k ‐curvature homogeneous if for any two points there exists a linear homothety between the corresponding tangent spaces which preserves the (1, 3)‐curvature operator and its covariant derivatives up to order k .…”

“…This concept lies between the notion of affine k ‐curvature homogeneity and k ‐curvature homogeneity since the group of homotheties lies between the orthogonal group and the general linear group. It was shown in that the existence of a linear homothety (i.e., such that is equivalent to the existence of a linear isometry and so . Hence, is Kowalski‐Vanžurová k ‐curvature homogeneous if for any two points there exists a linear isometry between the corresponding tangent spaces such that or, equivalently, if there are constants and so that for every point p of M , there is a basis for …”

Algebraic restrictions on geometric realizations of curvature models

“…See [9] for more information concerning weak curvature homogeneity. Homothety curvature homogeneity originated with the work in [13] and then subsequently in [14]; see also [5,6], and [4]. Our definition above is equivalent to the original definition given in [13], as was established in [5] or [6].…”