A common generalization of curvature homogeneity theories

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].…”

“…For these reasons, for i ≥ 2 we define A i ∈ ⊗ 4+i V * to be an algebraic curvature tensor 1 if it and its associated operator satisfy the relation in Equation (2.i). In addition, these tensors must also be antisymmetric in the first and second slots, be symmetric in the (1,2) and (3,4) slots, and the cyclic sum in slots one through three (first Bianchi identity) and slots three through five (second Bianchi identity) must be zero. If i = 0 or 1, then A i must satisfy Equations (2.g) or (2.h).…”

“…The most basic of these is the classical fact that, up to local isometry, there is a unique manifold (a space form) that geometrically realizes a 0-model of constant sectional curvature. Other examples of this study includes [2,7,8,12,17,18], and very recently, [4].…”

“…This condition has yet to be well-studied, however it seems to originate in[6]. See also what could be a collection of hypotheses in what is called variable curvature homogeneity as defined in Definition 1.1.1 of[4].…”