A thin homotopy between paths $f,g: I \to X$ in a topological space$X$ (with $I = [0,1]$ the standard interval) is a homotopy$I\times I \to X$ which, roughly speaking, has zero area.

Definition

Thin homotopies in smooth spaces

A (smooth) homotopy$F: I \times I \to X$ between smooth paths in a smooth space$X$ is called thin if the rank of its differential $d F(s,t): T_{s,t} I \times I \to T_{F(s),F(t)} X$ is less than 2 for all $s,t \in I \times I$.