Constant curvature, space of

A Riemannian space $M$ for which the sectional curvature $K(\sigma)$ is constant in all two-dimensional directions $\sigma$; if $K(\sigma)=k$, then the space is said to be of constant curvature $k$. By Schur's theorem, a Riemannian space $M^n$, $n>2$, is a space of constant curvature if for any point $p\in M$ the sectional curvatures $K(\sigma)$ in the directions of every two-dimensional subspace $\sigma$ of the tangent space $T_pM$ are the same. The curvature tensor of a space of constant curvature is expressed in terms of the curvature $k$ and the metric tensor $g_{ij}$ by the formula

$$R_{jlk}^i=k(\delta_k^ig_{jl}-\delta_l^ig_{jk}).$$

A space of a constant curvature is a locally symmetric space.

Up to an isometry there exists a unique complete simply-connected $n$-dimensional Riemannian space $S^n(k)$ of constant curvature $k$. For $k=0$ it is Euclidean space, for $k>0$ it is the sphere of radius $1/\sqrt k$, for $k<0$ it is Lobachevskii space.

The spaces $S^n(k)$ are maximal homogeneous spaces, i.e. their group of motions has maximum possible dimension $n(n+1)/2$. All maximal homogeneous Riemannian spaces different from $S^n(k)$ are exhausted by projective (elliptic) spaces obtained from spheres by identification of antipodal points.

Complete but multiply-connected spaces of constant curvature are called space forms. They are obtained by factorizing a simply-connected space $S^n(k)$ by a freely-acting discrete group of motions of $S^n(k)$. All space forms of positive curvature are known. The problem of classifying space forms of zero curvature and negative curvature has not yet (1983) been completely solved.

Spaces of constant curvature are distinguished from the other Riemannian spaces by one of the following characteristic properties: 1) spaces of constant curvature satisfy the axiom of planes, i.e. through every point and in the direction of every plane element at this point there passes a totally-geodesic submanifold; and 2) a space of constant curvature is a locally projectively-flat space, i.e. it admits locally projective mappings into Euclidean space.

The notion of a space of constant curvature does not have the property of "well-posedness" ("correctness"): a space with slowly varying sectional curvatures may be very different from a space of constant curvature. However, certain common properties of spaces of constant curvature, for example the topological structure, are preserved (the Hadamard–Cartan theorem, the sphere theorem, etc., see Curvature, [2]). In the class of pseudo-Riemannian spaces of constant curvature the situation is completely different: Any pseudo-Riemannian space of dimension exceeding 2 and with sectional curvature of fixed sign is a space of constant curvature.

Spaces of constant curvature are also locally conformally flat, i.e. they admit locally conformal mappings into Euclidean space.

References

[1]	J.A. Wolf, "Spaces of constant curvature" , Publish or Perish (1977)
[2]	Yu.D. Burago, V.A. Zalgaller, "Convex sets in Riemannian spaces of non-negative curvature" Russian Math. Surveys , 32 (1977) pp. 1–57 Uspekhi Mat. Nauk , 32 : 3 (1977) pp. 3–55

Comments

A Riemannian manifold of constant curvature is said to be elliptic, hyperbolic or flat according as the sectional curvature is positive, negative or zero. References [a1] and [1] contain a proof of Schur's theorem and give explicit constant curvature metrics. The classification of the compact spaces of constant positive curvature was done by J.A. Wolf. A Riemannian space $M$ is a locally symmetric space if $\nabla R=0$, cf. also Symmetric space.

References

[a1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. V, VI