an Einstein manifold
Among all Riemannian metrics, some are especially balanced: their Ricci curvature is the same in every direction, proportional to the metric itself. These are the Einstein manifolds. The name honors Einstein's gravity, where the vacuum field equations (with a cosmological constant) say exactly this — that the Ricci tensor is a constant multiple of the metric — so Einstein manifolds are the geometric models of a homogeneous, source-free gravitational universe.
A Riemannian manifold (M, g) is Einstein if Ric = lambda g for some constant lambda, i.e. the Ricci tensor is everywhere a fixed scalar times the metric. Taking the trace gives lambda = S/n where S is the scalar curvature and n the dimension, so an Einstein manifold automatically has CONSTANT scalar curvature. In dimensions n >= 3 a beautiful rigidity holds via the second Bianchi identity (Schur's lemma): if Ric = f g for a function f, then f is forced to be constant, so you only need the pointwise proportionality, not the constancy, to conclude the manifold is Einstein. Dimensions 2 and 3 are degenerate: every surface is trivially Einstein, and in dimension 3 Einstein already forces constant sectional curvature.
Where it matters and an honest caveat. Einstein metrics are the critical points of the total scalar curvature functional, central in geometry and physics, and finding them on a given manifold is a hard PDE problem. The crucial honesty: 'Einstein' constrains only the Ricci (trace) part of curvature, so in dimension n >= 4 an Einstein manifold is generally NOT of constant sectional curvature — the trace-free Weyl curvature is unconstrained. Constant-curvature space forms are Einstein, but the converse fails badly: K3 surfaces and many others are Einstein without being space forms.
The round sphere, flat space, and hyperbolic space are all Einstein (with lambda positive, zero, negative). But so is complex projective space with its Fubini-Study metric and the K3 surface with its Calabi-Yau Ricci-flat metric (lambda = 0) — and these are decidedly NOT constant-curvature, showing Einstein is strictly weaker than space form in dimension >= 4.
Space forms are Einstein, but Einstein is strictly more general: K3 and CP^n are Einstein without constant curvature.
Einstein constrains only the Ricci/trace part; the Weyl curvature is free. So in dimension >= 4, Einstein does NOT mean constant sectional curvature — that stronger condition is what defines a space form.