  We propose a topological Euler number to characterize nontrivial topological
phases of gapped fermionic systems, which originates from the Gauss-Bonnet
theorem on the Riemannian structure of Bloch states established by the real
part of the quantum geometric tensor in momentum space. Meanwhile, the
imaginary part of the geometric tensor corresponds to the Berry curvature which
leads to the Chern number characterization. We discuss the topological numbers
induced by the geometric tensor analytically in a general two-band model. As an
example, we show that the zero-temperature phase diagram of a transverse field
XY spin chain can be distinguished by the Euler characteristic number of the
Bloch states manifold in a (1+1)-dimensional Bloch momentum space.
