Weyl tensor

Remarks

Example: the unit sphere $S^n$, $n \ge 3$, has $R(X, Y)Z = g(Y, Z)X - g(X, Z)Y$, so $\operatorname{Rm} = \tfrac{1}{2} \, g \odot g$, $\operatorname{Ric} = (n - 1)g$ and $S = n(n - 1)$. Then $P = \tfrac{1}{2} g$ and $P \odot g = \tfrac{1}{2} \, g \odot g = \operatorname{Rm}$, so $W = 0$.

Non-example: for $n = 2$ the Schouten tensor has the factor $1/(n - 2)$ and is not defined, so surfaces have no Weyl tensor in this sense.

The Weyl tensor has the symmetries of the curvature tensor and all its traces vanish. For $n = 3$ it is always zero. For $n \ge 4$, $W = 0$ if and only if each point has a neighbourhood on which $g$ is a positive function times a flat metric. Under a change $\tilde g = e^{2f} g$, the Weyl tensor changes by $\tilde W = e^{2f} W$. Sign and index conventions for $\operatorname{Rm}$ differ between texts, and the formula for $W$ changes accordingly.