  Complex and spinorial techniques of general relativity are used to determine
all the states of the $SU(2)$ invariant quantum mechanical systems in which the
equality holds in the uncertainty relations for the components of the angular
momentum vector operator in two given directions. The expectation values depend
on a discrete quantum number and two parameters, one of them is the angle
between the two angular momentum components and the other is the quotient of
the two standard deviations. Allowing the angle between the two angular
momentum components to be arbitrary, \emph{a new genuine quantum mechanical
phenomenon emerges}: It is shown that although the standard deviations change
continuously, one of the expectation values changes \emph{discontinuously} on
this parameter space. Since physically neither of the angular momentum
components is distinguished over the other, this discontinuity suggests that
the genuine parameter space must be a \emph{double cover} of this classical
one: It must be diffeomorphic to a \emph{Riemann surface} known in connection
with the complex function $\sqrt{z}$. Moreover, the angle between the angular
momentum components plays the role of the parameter of an interpolation between
the continuous range of the expectation values found in the special case of the
orthogonal angular momentum components and the discrete point spectrum of one
angular momentum component. The consequences in the \emph{simultaneous}
measurements of these angular momentum components are also discussed briefly.
