# Geometrical interpretation of curvature invariants

Consider a Riemannian manifold. It is possible to describe it by curvature invariants. Now, is there any geometrical description (intuition) for simple invariants such as scalar curvature, Ricci square, Kretschmann and Weyl square?

In $$n$$ dimensions, the Ricci scalar curvature $$R$$ at a point measures how much the volume $$V$$ of a small $$n$$-dimensional ball of radius $$\epsilon$$ around that point differs from the Euclidean value $$V_E$$:
$$\frac{V}{V_E}=1-\frac{R}{6(n+2)}\epsilon^2+O(\epsilon^4).$$