Adriano,
One of the unique characteristics of the Luftwaffe (as opposed to independent air forces of other nations), was the possession of an organic paratrooper force; the Fallschirmjäger. These were established in 1938. So, I guess the answer to your, query, and the reason that you apparently found BAUER, Wilhelm. (DOB: 08.05.12). 27.05.41 Oblt., KIA Crete. in the Luftwaffe Officer Career Summaries here
http://ww2.dk/lwoffz.html , is that Gebirgsjäger officers were nominally the Luftwaffe officers.
Regards,
Sinisa