The Torus vs Its Embedding in Three-Space
July 3, 2016
As a visual person, this passage from Geometry, Topology and Physics by Nakahara Nakahara, Mikio. Geometry, Topology and Physics. CRC Press, 2003 blew my mind:
The Torus is a product manifold of two circles . If we denote the polar angle of each circle as , the coordinates of are . Since each is embedded in , may be embedded in . We often imagine as the surface of a doughnut in , in which case we inevitable have to introduce bending of the surface. This is an extrinsic feature brought by the ‘embedding’. When we say ‘a torus is a flat manifold’, we refer to the flat surface embedded in .
The bending of the torus is an extrinsic feature brought by the embedding! This makes a lot of sense: I couldn’t subconsciously associate the product of two circles with the torus in 3D, because they are in fact different: the 3D torus is an embedding of the cartesian product of two circles in 3D, not the object itself.
Another reminder that visualizing things, while helpful, can be misleading.