Kretschmann scalar

Remarks

The value of $K$ at a point does not depend on the coordinates, since it is a full contraction of tensors.

Example: for the Schwarzschild metric of mass $M$, in units $G = c = 1$, a computation of the curvature components gives $$K = \frac{48 M^2}{r^6}.$$ At the horizon $r = 2M$ this is $3/(4M^4)$, which is finite, while $K \to \infty$ as $r \to 0$.

Non-example: the Ricci scalar $R = g^{ac} g^{bd} R_{abcd}$ is a different invariant. For the Schwarzschild metric it is $0$ everywhere, so it does not detect the curvature that $K$ detects.

In a Riemannian manifold $K \ge 0$, and $K = 0$ exactly where the curvature tensor vanishes. In a Lorentzian manifold $K$ can vanish where the curvature does not, for example in plane gravitational wave spacetimes.