A virtual holonomic constraint is a relation among the coordinates of a mechanical system that can be made invariant via feedback control. This paper frames virtual holonomic constraints in an affine geometry setting, describing the dynamics on the constraint as the geodesics of a connection (the induced connection) derived from the Levi-Civita connection of the mechanical system. It also presents conditions for the constraint dynamics to be Lagrangian, based on the metrizability properties of the induced connection.