When I initiate Death from Above, does the Jump check trigger an attack of opportunity?
Jump skill checks don't provoke attacks of opportunity; the movement the creature takes as a result of the the jump, however, can provoke attacks of opportunity. Pedantry aside, if what you mean is Does the movement that's part of the maneuver provoke attacks of opportunity? I'd argue...
Death from Above's Movement Doesn't Provoke
The 4th-level Tiger Claw maneuver death from above (strike) reads, in part,
Emphasis mine. The maneuver death from above abstracts precise positioning and trajectory as part of the maneuver's effect, making it impossible to determine exactly which creatures (if any) should get attacks of opportunity versus the initiator.
I suspect that the intent is to permit an attack while the target is flat-footed then allow the attacker to safely get some distance from the target, predicated on a fairly easy skill check, not to have the initiator make the Jump check only for the foe to stab the initiator in mid-leap while attempting to soar over his foe.
A legalistic DM may require the initiator of the maneuver death from above to trace his path up the adjacent creature and then up to 20 ft. away and allow creatures along that path attacks of opportunity if the initiator doesn't somehow avoid them (e.g. a Tumble skill check). However, given that the adjacent creature could be a Colossal primal elemental or a Tiny grig, that first seems unfair, and given that there's no limit on how much travel the initiator can do before he lands up to 20 ft. away, the initiator should feel free to visit the moon, circle the earth, and then return in a safe position, probably avoiding at least some attacks of opportunity.
It'll ultimately be up to the DM if the movement from the maneuver death from above provokes attacks of opportunity, but the DM'll probably agree to let the attacks of opportunity slide if the initiator agrees not to visit outer space.
According To the Player's Handbook:
So any form of movement that brings the player outside a threatened square provokes an attack of opportunity unless specified otherwise. As Hey I Can Chan points out, the description implies that the user never leaves the threatened squares to make the attack.
*I'm Interpreting the word "Usually" as "unless specified otherwise" (E.G. Tumble)
Death from above dose not threaten Attacks of Opt. from the target because he no longer threatens him as he is considered flat-footed, however the adept must suffer Attacks of Opt. from any other enemy that would be threatening him at the time of this maneuver.