The Born-Oppenheimer electronic wave function Φ^BO_R (r) picks up a topological phase factor ±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^iγ if the geometric phase γ = φIm 〈 Φ_R | ∇_μ 〉 · dR_μ is evaluated with the conditional electronic wave function Φ_R (r) from the exact electron-nuclear factorization Φ_R (r) X (R) instead of the adiabatic function Φ^BO_R (r). A model of a pseudorotating triatomic molecule, also applicable to dynamical Jahn-Teller ions in bulk crystals, provides examples of nontrivial induced vector potentials and molecular geometric phase from the exact factorization. The induced vector potential gives a contribution to the circulating nuclear current that 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 wave function.