Geometric invariants as detector of Hawking and Unruh effects and quantum field theory in curved space

We report on the recent results revealing the presence of geometric invariants in all the phenomena in which vacuum condensates appear and we show that Aharonov--Anandan phase can be used to provide the evidence of phenomena like Hawking and Unruh effects and to test some behavior of quantum field theory in curved space. A very precise quantum thermometer can be also built by using geometric invariants.