  Geometry of the wave function is a central pillar of various geometric and
topological phenomena. Especially, in topological phases of matter, the wave
function geometry is expected to manifest in the form of various topological
invariants as well as anomalous topological responses. In this work, we unveil
the wave function geometry of two-dimensional semimetals with band crossing
points (BCPs). We show that the geometric properties of BCPs are governed by
the quantum metric describing the infinitesimal distance between quantum
states. For generic linear BCPs (LBCPs), we show that the corresponding
quantized Berry phase is determined either by an angular integral of the
quantum metric or, equivalently, by the maximum quantum distance of Bloch
states, which naturally explains the origin of the $\pi$-Berry phase of a LBCP.
In the case of quadratic BCPs (QBCPs), the path-independent Berry phase can
take an arbitrary value between 0 and $2\pi$. We find simple relations between
the geometric property of QBCPs and the quantum metric in two cases: i) when
one of the two crossing bands is flat, ii) when the system has rotation and/or
time-reversal symmetries. We show that, when the Berry curvature is absent, a
QBCP with an arbitrary Berry phase always accompanies another QBCP so that the
fermion-doubling theorem is satisfied. On the other hand, when finite Berry
curvature exists, a single QBCP with an arbitrary Berry phase can exist in the
Brillouin zone. This work demonstrates that the quantum metric plays a critical
role in understanding the fundamental properties of topological semimetals.
