  Two-band Chern insulators are topologically classified by the Chern number,
$c$, which is given by the integral of the Berry curvature of the occupied band
over the Brillouin torus. The curvature itself comes from the imaginary part of
a more basic object, the quantum geometric tensor, $Q$. On the other hand, the
integral over the Brillouin torus of the real part of $Q$ gives rise to another
magnitude, the quantum volume, $v_{g}$, that like $c$, jumps when the system
undergoes a topological phase transition and satisfies the inequality $v_{g}\ge
2\vert c \vert$. The information contained in $v_g$ about the topology of the
system has been investigated recently. In this paper we present new results
regarding the underlying geometric structure of two-dimensional two-band
topological insulators. Since a generic model describing the system can be
characterized by a map, the classifying map, from the Brillouin torus to the
two-sphere, we study its properties at the geometric level. We present a
procedure for splitting the Brillouin torus into different sectors in such a
way that the classifying map when restricted to each of them is a local
diffeomorphism. By doing so, in the topological phases we are able to isolate a
region contained in the Brillouin torus whose volume is the minimal one,
$v_{min}=2\vert c \vert$ and the integral of the Berry curvature on it is $c$.
For cases in which $v_{g}> 2\vert c \vert$, the regions contributing to the
excess of volume, $v_{ex}=v_{g}-2\vert c \vert$, are found and characterized.
In addition, the present work makes contact with, and clarifies, some
interpretations of the quantum volume in terms of the Euler characteristic
number that were done in the recent literature. We illustrate our findings with
a careful analysis of some selected models for Chern insulators corresponding
to tight-binding Hamiltonians.
