Does $R(X, Y)Y$ lie in the plane spanned by $X$ and $Y$? [closed]

Let $X$, $Y$ be tangent vectors to a point on a Riemannian manifold $(M, g)$. Clearly $R(X,Y)Y$ is orthogonal to $Y$. Is it nonetheless the case that $R(X,Y)Y$ lies in the span of $X$ and $Y$? If so, how is this proved? If not can you give a counterexample?

Solution 1:

If what you say is true for a manifold for any $X$ and $Y$ then, by taking $X$ orthogonal to $Y$, $R(X,Y)Y$, being orthogonal to $Y$, must be a multiple of $X$. Fixing a point and summing over a orthogonal basis of $\{X\}^{\perp}$, one gets that the Ricci operator of $X$ is a multiple of $X$. So $\operatorname{Ric}_p$ has every vector of $T_pM$ as eigenvectors. By linear algebra this implies that $\operatorname{Ric}_p=\lambda(p)g_p$, and since this is valid for every point, $g$ must be an Einstein metric. So any non Einstein metric is a counterexample.