Constant curvature riemann manifold

Question

So, I am studying cosmology, and while investing the math involved with the assumptions regarding the metric used (isotropic and homogeneous), I stumbled upon a definition that was recurring, that of a constant curvature manifold (don't know if it's the right terminology).

If I'm not mistaken, the definition is any manifold such that it's riemann tensor has the following form :

$$ R_{abcd} = K(g_{ac}g_{bd}-g_{ad}g_{bc})$$

We notice that for a riemann tensor of this form, the Ricci scalar is indeed constant and proportional to K.

My question is : is the definition of a constant curvature manifold (or whatever the right object is, if I remember correctly a manifold isn't necessarily endowed with a metric) the one I gave on $R_{abcd}$ ?

It would be more intuitive for me if constant curvature was simply a manifold with constant Ricci scalar. If this is indeed the definition, is there an easy way to prove the general form of $R_{abcd}$ ?

Answer 1

In your setting, "constant curvature" means constant sectional curvature. The usual definition is "every tangent $2$-plane has the same curvature"; the formula you give for the Riemann curvature tensor is then a theorem.

In differential geometry, a metric with constant Ricci curvature is usually called an Einstein metric.

Being Einstein is not equivalent to having constant curvature. Up to overall scaling, there are precisely three local models for a space of constant curvature (and three complete, simply-connected examples: the sphere, Euclidean space, and hyperbolic space). By contrast, there are moduli spaces of Einstein metrics even on a single smooth manifold. For example, the smooth $4$-manifold underlying a complex quartic in complex projective $3$-space admits, up to scaling, a $20$-dimensional family of Ricci-flat Kähler metrics by Yau's solution of the Calabi conjecture.