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. (read more)