  The Higgs potential of 2HDM keeps its generic form under the group of
transformation GL(2,C), which is larger than the usually considered
reparametrization group U(2). This reparametrization symmetry induces the
Minkowski space structure in the orbit space of 2HDM. Exploiting this property,
we present a geometric analysis of the number and properties of stationary
points of the most general 2HDM potential. In particular, we prove that
charge-breaking and neutral vacua never coexist in 2HDM and establish
conditions when the most general explicitly CP-conserving Higgs potential has
spontaneously CP-violating minima. Our analysis avoids manipulation with
high-order algebraic equations.
