  In the present paper, we propose a new way to classify centrosymmetric metals
by studying the Zeeman effect caused by an external magnetic field described by
the momentum dependent g-factor tensor on the Fermi surfaces. Nontrivial U(1)
Berry's phase and curvature can be generated once the otherwise degenerate
Fermi surfaces are splitted by the Zeeman effect, which will be determined by
both the intrinsic band structure and the structure of g-factor tensor on the
manifold of the Fermi surfaces. Such Zeeman effect generated Berry's phase and
curvature can lead to three important experimental effects, modification of
spin-zero effect, Zeeman effect induced Fermi surface Chern number and the
in-plane anomalous Hall effect. By first principle calculations, we study all
these effects on two typical material, ZrTe$_5$ and TaAs$_2$ and the results
are in good agreement with the existing experiments.
