  Heisenberg and Schr{\"o}dinger uncertainty principles give lower bounds for
the product of variances $Var_{\rho}(A)\cdot Var_{\rho}(B)$, in a state $\rho$,
if the observables $A,B$ are not compatible, namely if the commutator $[A,B]$
is not zero.
  In this paper we prove an uncertainty principle in Schr{\"o}dinger form where
the bound for the product of variances $Var_{\rho}(A)\cdot Var_{\rho}(B)$
depends on the area spanned by the commutators $[\rho,A]$ and $[\rho,B]$ with
respect to an arbitrary quantum version of the Fisher information.
