Parametrized homotopic distance

We introduce the concept of parametrized homotopic distance, extending the classical notion of homotopic distance to the fibrewise setting. We establish its correspondence with the fibrewise sectional category of a specific fibrewise fibration and derive cohomological lower bounds and connectivity upper bounds under mild conditions. We also analyze the behavior of parametrized homotopic distance under compositions and products of fibrewise maps, along with its interaction with the triangle inequality. We establish several sufficient conditions for fibrewise $H$-spaces to admit a fibrewise division map and prove that their parametrized topological complexity equals their fibrewise unpointed Lusternik-Schnirelman category, extending Lupton and Scherer's theorem to the fibrewise setting. Additionally, we give sharp estimates for the parametrized topological complexity of a class fibrewise $H$-spaces which arises as sphere bundles with fibre $S^7$. Furthermore, we estimate the parametrized homotopic distance of fibre-preserving, fibrewise maps between fibrewise fibrations, in terms of the parametrized homotopic distance of the induced fibrewise maps between individual fibres, as well as the fibrewise unpointed Lusternik-Schnirelman category of the base space. Finally, we define and study a pointed version of parametrized homotopic distance, establishing cohomological bounds and identifying key conditions for its equivalence with the unpointed version, thus providing a finer classification of fibrewise homotopy invariants.