Geometric realizations of curvature models by manifolds with constant scalar curvature

Abstract

We show any Riemannian curvature model can be geometrically realized by a manifold with constant scalar curvature. We also show that any pseudo-Hermitian curvature model, para-Hermitian curvature model, hyperpseudo-Hermitian curvature model, or hyper-para-Hermitian curvature model can be realized by a manifold with constant scalar and ⋆-scalar curvature.

Geometric Realizations of Para-Hermitian Curvature Models

“…The Weyl conformal curvature tensor W := A − σρ is the projection of A on ker(ρ); we say a model or a pseudo Riemannian manifold is conformally flat if and only if W = 0. The following result [11] shows, in particular, that the relations of Equation (3.a) generate the universal symmetries of the Riemann curvature tensor. We focus our attention on the scalar curvature:…”

Geodesically equivalent metrics in general relativity

“…Thus, if (Γ, f ) is a solution of (6), then for every 1−form φ the pair Γ ,f given by (8,9) is also a solution. Let us show that up to this gauge freedom the connection Γ and the function f are unique.…”

Geometric realizations, curvature decompositions, and Weyl manifolds

“…Let V be a real vector space of dimension n ≥ 2 which is equipped with a non-degenerate symmetric inner product h. Let R(V ) ⊂ ⊗ 4 V * be the space of all generalized curvature tensors; A ∈ R(V ) if and only if A satisfies the symmetries given in Equations (2) and (3). The space of algebraic curvature tensors A(V ) ⊂ R(V ) is the subspace defined by imposing in addition the symmetry of Equation (5). An immediate algebraic consequence of Equations (2), (3), and (5) is the additional symmetry A(x, y, z, w) = A(z, w, x, y) .…”

“…If (N, g, ∇) geometrically realizes A at a point P ∈ N , by considering a suitable conformal deformation (N, e 2f g, ∇), we can use the Cauchy-Kovalevskaya Theorem to construct a Weyl manifold where f = O(|x − P | 3 ) which has constant scalar curvature and which realizes A at P . The argument is essentially the same as that used in [5] to establish a similar fact in the pseudo-Riemannian setting so we omit details in the interests of brevity.…”