Abstract: The real and imaginary parts of quantum geometric tensor are of great significance and they can help us to understand the geometric and topological properties of quantum systems clearly. In this paper, from the case of gauge transformation acting on the real space and then extending it to an abstract parametric space, the tensors of quantum geometry with their relevant concepts are introduced in detail, which enable us a further understanding and a deep recognization of quantum geometry for quantum applications.

Key words: gauge transformation, quantum geometric tensor, quantum metric tensor, Berry curvature