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Brownian motion is a universal characteristic of colloidal particles embeddedin a host medium, and it is the fingerprint of molecular transport ordiffusion, a generic feature of relevance not only in Physics but also inseveral branches of Science and Engineering. Since its discovery, Brownianmotion or colloid dynamics has been important in elucidating the connectionbetween the molecular details of the diffusing macromolecule and themacroscopic information of the host medium. However, colloid dynamics is farfrom being completely understood. For example, the diffusion of non-sphericalcolloids and the effects of geometry on the dynamics of either passive oractive colloids are a few representative cases that are part of the currentchallenges in Soft Matter Physics. In this contribution, we take a step forwardto introduce a covariant description of the colloid dynamics in curved spaces.This formalism will allow us to understand several phenomena, for instance, theeffects of curvature on the kinetics during spinodal decomposition and thethermodynamic properties of the colloidal dispersion, just to mention a fewexamples. This theoretical framework will also serve as the starting point tohighlight the role of geometry on colloid dynamics, an aspect that is ofparamount importance to understanding more complex phenomena, such as thediffusive mechanisms of proteins embedded in cell membranes.

Covariant description of the colloidal dynamics on curved manifolds