This place is filled with Cie’th and so your battles here will be against Cie’th-type enemies. There is a split path here but it just loops round in a circle where they merge again. It doesn’t matter which route you take but you should fight all of the enemies here for CP by running the loop round.
Strigoi are very weak to elemental skills and can be easily Staggered so make good use of Ravagers when fighting them. A good tactic is to debuff them first with a SAB and then switch to Relentless Assault to finish them off. They are not particularly dangerous and you will only fight one at a time here. Kill off the six of them in this little section then follow the path onwards.
After the paths merge, there is a Treasure Sphere floating right in front of you which contains Perovskite . A different kind of Cie’th awaits just ahead in the form of a Vetala - They are immune to physical attacks until Staggered so use of Ravagers is a must when trying to deal with them. Once Staggered, you can inflict Deprotect and switch to all COMs to finish it off. Magic is also rather ineffective and should just be used to Stagger the Vetala.
Another couple of Strigoi are encountered just past the Vetala which you will have to fight together but the same rules apply. Just be a little more cautious and you shouldn’t have anything to worry about. Finally, the door past the Save Station can be opened but you need to have accepted a Cie’th Stone mission in order to make it budge. There is nothing more to do here so return back to Ceiling of Sky (where you were before you boarded Atomos a second time).