Constant sectional curvature and space form

This item assumes ${\mathrm{A}\mathrm{C}}_{\omega }$, namely countable choice. In the propagated dependency chain, that assumption is required through Sectional curvature; after those interfaces are fixed, the remaining local or finite argument makes no additional countable-family choice.

Assume the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ exactly as in Geodesically complete Riemannian manifold. It is used here only through that supplier's construction of the unique maximal geodesic domains needed to interpret completeness; the constant-curvature predicate itself makes no further family choice beyond the stated inherited assumption.

A Riemannian manifold has constant sectional curvature $K\in \mathbb{R}$ when every tangent two-plane at every point has sectional curvature $K$. A connected, boundaryless, geodesically complete Riemannian manifold of constant sectional curvature is called a space form on this page.

This global condition is stronger than saying that, at each point $p$, all two-planes have some common value $K(p)$; Schur's lemma later proves constancy of that pointwise value in connected dimension at least three. In dimensions zero and one there are no tangent two-planes, so the predicate “has constant sectional curvature $K$” is vacuous for every $K$ and does not determine a distinguished number. Under the library convention that the empty space is connected, the empty boundaryless complete manifold is correspondingly a vacuous space form. These low-dimensional conventions do not affect later formulas, whose alternating metric model vanishes in dimensions below two.