I have a curve that looks like the one shown in the figure below.
The curve is defined on a torus. I would like to compute the integral of a two form over the area enclosed by this curve. I understand using Stokes' theorem I can get the line integral of the corresponding one form along the boundary, however, I would like to stick to integrating the two form.
What is the area enclosed that I should consider?
Some closed curves are the boundaries of areas and others are not. The number of different curves that are not boundaries is measured by the first homology group of the space. The torus has first homology group $\mathbb{Z}^2$, which means there are two independent curves that aren't the boundaries of areas. These two curves are precisely the (non-contractible) longitude and meridian loops. So the curve you've drawn is not the boundary of any area!