Questions regarding post-PhD research jobs (specifically mathematics)
Discalimer: IANADG, I am not a differential geometrist...

Still, some potentially-relevant fields:


Architecture and materials industry: Surfaces and bodies with complex structures, their behavior, the forces applied to them, their expansion, contortion etc. with weather or with load changes and so on. Specifically, I can imagine a bunch of relevant applications in 3d-printing. This might be relevant (although it's not a specific workplace)
Car industry, for similar reasons
Computer vision and image/video  processing: lots of geometry and lots of derivation going on there. Just as an example, read this (but )


I'll also make another, more concrete suggestion: Various countries have academic-oriented placement/recruitment agencies. While those obviously don't know you perfectly well and can't cater to your interests exactly, they could interest you in a selection of potential positions which you might find appealing - or, at least, will help you realize what kind of work is out there. Also, an up-to-date LinkedIn profile has a bit of that same effect.