Differential formulation of Schrodinger equation leads to vanishing Berry phase

The Poincare-Hopf theorem states that a globally smooth tangent vector does not exist on a manifold whose Euler characteristic is non-zero. Nevertheless, when one defines a differential equation on such a manifold, this theorem is always ignored. For example, the differential formulation of Schrodinger equation is defined as a form so that a tangent vector is a product of Hamiltonian and state vector. In this case, if the Hamiltonian and the state vector are both globally smooth functions on parameter space, then the tangent vector will be compelled to become smooth so that the Euler characteristic of the parameter space must be zero. As a result, some Berry phases related to non-zero Euler characteristic will be ruled out.