  Chern number is usually characterized by Berry curvature. Here, by
investigating the Dirac model of even-dimensional Chern insulator, we give the
general relation between Berry curvature and quantum metric, which indicates
that the Chern number can be encoded in quantum metric as well as the surface
area of the Brillouin zone on the hypersphere embedded in Euclidean parameter
space. We find that there is a corresponding relationship between the quantum
metric and the metric on such hypersphere. We show the geometrical property of
quantum metric. Besides, we give a protocol to measure the quantum metric in
the degenerate system.
