You may wish to add the additional condition that the areas must be equal.  Otherwise, you can make all but one region be very tiny.  Here's a physical solution.  Fill the manifold with N mutually immiscible liquids with equal surface tensions between the liquids.  Shake, and let surface tension minimize the total boundary.   --  Gene On Saturday, April 7, 2018, 2:46:25 PM PDT, Dan Asimov <dasimov@earthlink.net> wrote: Consider a smooth surface M like the unit sphere S^2 or the square torus T^2 = R^2/Z^2. Question: ----- For which integers N > 0 can M be divided into N regions of positive area such that the total length of the regions' boundaries is a local minimum? ----- The question becomes interesting when N is big enough for the surface to be divided into N *simply connected* regions. (E.g., the torus cannot be cut into fewer than N >= 4 simply connected pieces. In that case the total boundary length (counting each edge only once) seems to be sqrt(8).) Can we find the optimal configuration for M=S^2 and M=T^2 for N <= 10 (and N >= 4 if M=T^2). Optimal meaning that the total length of the boundary is a global minimum. For further study: What is the minimum number of simply connected pieces that the surface of genus g can be cut into?) —Dan