Let $(M,g)$ be a semi-Riemannian manifold. Let $\gamma : I \rightarrow M$ be a geodesic. Denote by $P= \frac{1}{n-2} \left( Ric- \frac{scal}{2(n-1)}g \right)$ the Schouten tensor. Does $P(\gamma'(t),X)=0$ hold for all $t \in I$ and all $X \in T_{\gamma(t)} M$?
The question arose after reading "Differential Geometry, Valencia 2001", edited by Gil-Medrano, Miquel, page 257, Proposition 2. (Link to Google Books) It seems the author uses a different notion of Schouten tensor there. Also "An extension theorem for conformal gauge singularities" by Tod, Lübbe, page 13, Proposition 2.1 (Link to Arxiv) seems to contradict the claim and rather suggest the following:
There exists a conformally equivalent metric $\tilde{g}$, such that $\gamma$ is a geodesic with respect to $\tilde{g}$ and the Schouten tensor $\tilde{P}$ induced by $\tilde{g}$ vanishes along $\gamma$, i.e. $\tilde{P}(\gamma'(t),X)=0$ for all $t \in I$ and all $X \in T_{\gamma(t)} M$.
Sadly I am not familiar with the methods used in either source, so I cannot really follow the proofs. I was hoping to receive the result with elementary methods. That means: So far I tried introducing a local, parallel orthonormal basis $(s_i)$ and directly compute $P(\gamma',s_i)$, but after expanding I can see no reason why the terms there should cancel out.