  Consider a set of quantum states $| \psi(x) \rangle$ parameterized by $x$
taken from some parameter space $M$. We demonstrate how all geometric
properties of this manifold of states are fully described by a scalar
gauge-invariant Bargmann invariant $P^{(3)}(x_1, x_2,
x_3)=\operatorname{tr}[P(x_1) P(x_2)P(x_3)]$, where $P(x) = |\psi(x)\rangle
\langle\psi(x)|$. Mathematically, $P(x)$ defines a map from $M$ to the complex
projective space $\mathbb{C}P^n$ and this map is uniquely determined by
$P^{(3)}(x_1,x_2,x_3)$ up to a symmetry transformation. The phase $\arg
P^{(3)}(x_1,x_2,x_3)$ can be used to compute the Berry phase for any closed
loop in $M$, however, as we prove, it contains other information that cannot be
determined from any Berry phase. When the arguments $x_i$ of
$P^{(3)}(x_1,x_2,x_3)$ are taken close to each other, to the leading order, it
reduces to the familiar Berry curvature $\omega$ and quantum metric $g$. We
show that higher orders in this expansion are functionally independent of
$\omega$ and $g$ and are related to the extrinsic properties of the map of $M$
into $\mathbb{C}P^n$ giving rise to new local gauge-invariant objects, such as
the fully symmetric 3-tensor $T$. Finally, we show how our results have
immediate applications to the modern theory of polarization, calculation of
electrical response to a modulated field and physics of flat bands.
