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.

The Torus vs Its Embedding in Three-Space - July 3, 2016 - Hang Lu Su