  The Born-Oppenheimer electronic wavefunction $\Phi_R^{BO}(r)$ picks up a
topological phase factor $\pm 1$, a special case of Berry phase, when it is
transported around a conical intersection of two adiabatic potential energy
surfaces in $R$-space. We show that this topological quantity reverts to a
geometric quantity $e^{i\gamma}$ if the geometric phase $\gamma = \oint
\mathrm{Im} \langle \Phi_R |\nabla_{\mu} \Phi_R\rangle \cdot d\mathbf{R}_{\mu}$
is evaluated with the conditional electronic wavefunction $\Phi_R(r)$ from the
exact electron-nuclear factorization $\Phi_R(r)\chi(R)$ instead of the
adiabatic function $\Phi_R^{BO}(r)$. A model of a pseudorotating molecule, also
applicable to dynamical Jahn-Teller ions in bulk crystals, provides the first
examples of induced vector potentials and molecular geometric phase from the
exact factorization. The induced vector potential gives a contribution to the
circulating nuclear current which cannot be removed by a gauge transformation.
The exact potential energy surface is calculated and found to contain a term
depending on the Fubini-Study metric for the conditional electronic
wavefunction.
