The constant-curvature spatial model of a homogeneous cosmology

This item assumes ${\mathrm{A}\mathrm{C}}_{\omega }$ as inherited through the Riemannian items cited below (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)); no further choice principle is used.

(a) The datum. Let $\mathrm{\Sigma }$ be a connected smooth three-manifold without boundary (Smooth manifolds and their smooth charts) and let $\gamma$ be a Riemannian metric on $\mathrm{\Sigma }$ (Riemannian metric and riemannian manifold). Let $k\in \{-1,0,1\}$. The pair $(\mathrm{\Sigma },\gamma )$ is a spatial model of constant normalized curvature $k$, with normalized curvature parameter $k$, when

$\mathrm{Ric}(\gamma )=2k\gamma \text{at every point of }\mathrm{\Sigma },$

where $\mathrm{Ric}(\gamma )$ is the Ricci curvature of $\gamma$ (Ricci curvature). We then call $(\mathrm{\Sigma },\gamma ,k)$ a spatial model datum. The signs are named for orientation only: $k=+1$ is called the closed or positively curved case, $k=0$ the flat case and $k=-1$ the open or negatively curved case; no topological content is attached to these names by this definition.

(b) Status of the condition. The condition in (a) is pointwise and local. It neither asserts nor implies that $\mathrm{\Sigma }$ is complete, compact, simply connected or geodesically complete, and it does not determine the global model: this definition proves no classification theorem and this page assumes none. In dimension three the condition is equivalent to the statement that $\gamma$ has constant sectional curvature $k$; the companion lemma on constant-curvature three-manifolds, recorded in the Remarks, proves that equivalence. What is used by the cosmological deductions of this page is exactly the Ricci form displayed in (a).

(c) Normalization and scale. The condition (a) fixes the numerical curvature to $k\in \{-1,0,1\}$. For a constant $\lambda >0$, put ${\gamma }^{\mathrm{\prime }}={\lambda }^2\gamma$. Then

$\mathrm{Ric}({\gamma }^{\mathrm{\prime }})=\mathrm{Ric}(\gamma )=2k\gamma =2(k\mathrm{/}{\lambda }^2){\gamma }^{\mathrm{\prime }},$

and $K_{{\gamma }^{\mathrm{\prime }}}=k\mathrm{/}{\lambda }^2$ (Constant metric rescaling preserves the connection and Ricci tensor and inversely scales scalar and sectional curvature). The sign is unchanged, but for $k\mathrm{\ne }0$ the normalized condition (a) is preserved only when $\lambda =1$; the flat condition is preserved for every $\lambda >0$. Conversely, if $\mathrm{Ric}(h)=2\kappa h$ for a real constant $\kappa \mathrm{\ne }0$, the metric $\gamma =\mathrm{\mid }\kappa \mathrm{\mid }h$ satisfies $\mathrm{Ric}(\gamma )=2\mathrm{sgn}(\kappa )\gamma$ by the same Ricci invariance, so it restores the normalization. A physical model assigns its spatial length scale through the further factor in the model postulate.

(d) Standard models. For each $k\in \{-1,0,1\}$ at least one complete, connected, boundaryless, simply connected model satisfying (a) is available:

• For $k=+1$, take the round unit sphere $S^3\subset {\mathbb{R}}^4$ with its induced metric. Its constant curvature $+1$, completeness and connected boundaryless Riemannian structure are supplied by Round sphere model geometry with $n=3$, $R=1$ (the curvature calculation is also The round sphere has positive constant sectional curvature); its simple connectedness is $S^n$ is simply connected for every $n\ge 2$.
• For $k=0$, take ${\mathbb{R}}^3$ with the standard metric. Its Riemannian structure and vanishing curvature are Euclidean space has zero curvature. It is nonempty and convex, hence simply connected by Every nonempty convex subset of ${\mathbb{R}}^n$ is simply connected, and thus connected by Every path-connected space is connected, and every path component lies inside a component. For completeness, the constant metric coefficients give zero Christoffel symbols (Christoffel formula for the levi civita connection); for every initial position $p$ and velocity $v$, the all-time curve $t\mapsto p+tv$ solves the coordinate geodesic equation (Coordinate geodesic equation). Uniqueness of the maximal geodesic (Existence uniqueness and smooth dependence of geodesics) identifies it with that curve, so every maximal geodesic has domain $\mathbb{R}$. Hopf–Rinow (Hopf–Rinow theorem) then gives metric completeness; its hypotheses hold because ${\mathbb{R}}^3$ is nonempty, connected and boundaryless.
• For $k=-1$, take the upper half-space $\{x\in {\mathbb{R}}^3:x^3>0\}$ with $y=x^3$ and metric $y^{-2}{\sum }_{a=1}^{3}d{x}^a\otimes d{x}^a$. Its complete, connected, boundaryless, simply connected Riemannian structure and constant curvature $-1$ are exactly the $n=3$, $a=1$ instance of Upper half-space model geometry.

By the companion lemma recorded in the Remarks, constant sectional curvature $k$ in dimension three gives $\mathrm{Ric}(\gamma )=2k\gamma$, so each model satisfies (a) and each sign is realized. A model datum restricts to one of these global standard models only when its postulate says so.

• Why a Ricci condition and not a metric formula. The displayed form $\mathrm{Ric}(\gamma )=2k\gamma$ is the exact input of the Einstein reduction performed later on this page: the warped-product computation of the four-dimensional Ricci tensor uses only $\mathrm{Ric}(\gamma )$, never the explicit coordinate form of a constant-curvature metric. The equivalence with constant sectional curvature in clause (b) is what allows the standard models of clause (d), and the physical reading of the parameter, to be described in the usual geometric language.

• Global model data. Clause (a) imposes the displayed Ricci condition. The global spatial manifold is a separate part of the datum, and no classification of all manifolds satisfying this condition is asserted. The physical postulate specifies which global model is adopted.

• Companion items on this page. The equivalence between the Ricci condition of clause (a) and constant sectional curvature in dimension three, used in clauses (b) and (d), is Constant-curvature three-manifolds: equivalence of constant sectional curvature, the tensor identity and the Ricci condition ↗. The invariance of the Ricci tensor under a constant rescaling of the metric, used in clause (c), is Constant metric rescaling preserves the connection and Ricci tensor and inversely scales scalar and sectional curvature.

• Choice convention. The cited Riemannian suppliers assume ${\mathrm{A}\mathrm{C}}_{\omega }$ through the construction of sectional curvature, geodesic domains and orthonormal frames; this definition inherits that assumption and uses no further choice principle.