On curvature‐homogeneous spaces of type (1,3)

Abstract

Key words Riemannian manifold, curvature homogeneous manifold, locally homogeneous space MSC (2010) 53C21, 53C25, 53C30 Curvature homogeneous spaces have been studied by many authors. In this paper, we introduce and study a natural modification of this class, namely so-called curvature homogeneous spaces of type (1,3). We present a class of proper examples in every dimension and we prove a classification theorem in dimension 3 (for the generic case). We restrict ourselves to the Riemannian situation and to curvature homogeneity of order zero.

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].…”

Homogeneity of Lorentzian three-manifolds with recurrent curvature

“…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 .…”

“…Remark 4.5. There is a different notion of curvature homogeneity that is due to Kowalski and Vanžurová [22], [23]. Motivated by their seminal work, we say that a manifold (M, g) 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 R 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.…”

Homothety curvature homogeneity and homothety homogeneity

“…There is a slightly different version of curvature homogeneity that we shall discuss here and which, motivated by the seminal work of Kowalski and Vanžurová [10,11], we shall call homothety k-curvature homogeneity. In Definition 1.1, we may replace the curvature tensor R by the curvature operator R since we are dealing with isometries.…”