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For example, I can create a "state space" isomorphic to R3 of TxPxV I can then define a surface on this space of PV=NkT I can define quasi static state equations as curves restricted to the surface.

Is this an approach a useful approach to thermodynamics? Is this studied at all? I Googled "Thermodynamics differential geometry" and got results that were about much more advanced topics in thermodynamics. Is there a text, or paper, studying basic (I am taking an undergrad course) thermodynamics from this perspective?