The invention relates to a composition comprising, in a physiologically acceptable medium containing a fatty phase, a polymer with a weight-average molecular mass of less than 100 000 and in particular ranging from 1 000 to 30 000, comprising a) a polymer skeleton containing hydrocarbon-based repeating units containing at least one hetero atom, and optionally b) pendent fatty chains and/or terminal fatty chains, which may be functionalized, containing from 6 to 120 carbon atoms and being linked to these units, and fibres.