  The eigenvalues of a parameter-dependent $N\times N$ Hamiltonian matrix form
a band structure in parameter space. Quantum geometric properties (Berry
curvature, quantum metric, etc.) of such $N$-band systems are usually computed
from parameter-dependent eigenstates. This approach faces several difficulties,
including gauge ambiguities and singularities in the multicomponent
eigenfunctions. In order to circumvent this problem, this work exposes an
alternative approach based on eigenprojectors and (generalized) Bloch vectors.
First, an expansion of each eigenprojector as a matrix polynomial in the
Hamiltonian is deduced, and using SU($N$) Gell-Mann matrices an equivalent
expansion of each Bloch vector is found. In a second step, expressions for the
$N$-band Berry curvature and quantum metric in terms of Bloch vectors are
obtained. This leads to new explicit Berry curvature formulas in terms of the
Hamiltonian vector, generalizing the well-known two-band formula to arbitrary
$N$. Moreover, a detailed treatment is given for the case of a particle-hole
symmetric energy spectrum, which occurs in systems with a chiral or charge
conjugation symmetry. For illustrating the formalism, several model
Hamiltonians featuring a multifold linear band crossing are discussed; they
have identical energy spectra but completely different geometric and
topological properties. The methodology used in this work is more broadly
applicable to compute any physical quantity, or to study the quantum dynamics
of any observable without the explicit construction of energy eigenstates.
