Prove that the linear map $X_m\mapsto(X_m\contract\O_m)^\sharp$ is skew symmetric, i.e. for all $X_m,Y_m\in T_mM$: $\la(X_m\contract\O_m)^\sharp,Y_m\ra=-\la X_m,(Y_m\contract\O_m)^\sharp\ra$. 2. If $Z_m\in T_mM$ is an instantaneous observer minimizing $X_m\mapsto\norm{X_m\contract\O_m)^\sharp}^2$, then $Z_m$ is an eigen-vector of
$$
X\mapsto((X\contract\O)^\sharp\contract\O)^\sharp~.
$$
1. What is the adjoint of $Z\mapsto(Z\contract\O)^\sharp$?
$$
\la(Z\contract\O)^\sharp,W\ra
=\O(Z,W)
=-\O(W,Z)
=-\la Z,(W\contract\O)^\sharp\ra
$$
Thus the adjoint is $W\mapsto-(W\contract\O)^\sharp$.
2. Suppose a time-like unit vector $Z$ minimizes $\la Z\contract\O,Z\contract\O\ra$. With Lagrange multiplier $\l$ we get the functional:
$$
Z\mapsto\la Z\contract\O,Z\contract\O\ra-\l(\la Z,Z\ra+1)
$$
Thus for all $N\in T_mM$:
$$
\la N\contract\O,Z\contract\O\ra-\l\la Z,N\ra=0,
\la Z,Z\ra=-1
$$
By the first part we have
$$
\la N\contract\O,Z\contract\O\ra
=-\la N,(Z\contract\O)^\sharp\contract\O\ra
$$
and therefore
$$
-((Z\contract\O)^\sharp\contract\O)^\sharp=\l Z
$$