Surfaces (Pt. II)
Point of Post: This is a continuation of this post.
So, we have proven that the unit sphere is a surface. That said, it should be intuitively obvious that any sphere of any radius and any center should be a surface. In fact, it should be the case that any ‘smooth’ deformation of into an ellipsoid, or something of the sort, should also be a surface. The operative term in this last sentence was ‘smooth’. Indeed, it’s evident that if one has a surface which can be thought of, at least locally, as the smooth deformation of , then evidently applying a smooth deformation to will result in something that locally is just a smooth deformation of –namely each point locally looks like you took smoothly deformed it into a piece of , and then smoothly deformed that piece of into the piece of . In other words, it seems intuitive that if we compose chart maps for with we should get maps that deform smoothly into pieces of . This intuition is formalized by the following theorem:
Theorem: Let be open. Suppose that is a diffeomorphism (i.e. a bijective function with inverse). If is a surface then is a surface.
Proof: Let be arbitrary. Since is a surface we may choose a smooth chart with . Consider then the map . Obviously is open in (since is a homeomorphism), is a homeomorphism (since they are both homeomorphisms), is smooth (since they’re both smooth) and . Thus, if we can verify that is injective for all we’ll be done. That said, from the chain rule we know that and since and are both injective ( by assumption, and (actually is invertible) by assumption that is a diffeomorphism) we may conclude that is injective. The conclusion follows.
This tells us that, in fact we were correct, that things like ellipsoids and spheres of arbitrary radius and center are surfaces, being just images of under affine transformations.
So, we have shown our one claim in the motivation, namely that the graph of a smooth function is a surface. We now show the other claim about level sets of curves. For the setup of this theorem we recall that we call, for a function where is open, the point a regular value if for every with one has that is non-zero, which, recalling that , is equivalent to . With this in mind we can state the following:
Theorem: Let be open and a smooth function. Then, if is a regular value for one has that is a surface.
Proof: Let be arbitrary. Since is a regular point for we know from the implicit function theorem that since we may find neighborhood of and open sets , with a smooth function with (the graph) from where we may easily conclude by our previous comments about graphs that has a differentiable chart at . Since was arbitrary the conclusion follows.
Said roughly, we used the implicit function theorem to say that every surface which is the level set of a smooth function at a regular point, locally looks like the graph of a function and thus is a surface.
This gives us convenient ways of proving a lot of objects are surfaces. For example:
Example: Let’s now show that the torus with tube radius , and distance from center of tube to center of hole equal to with is a surface (centered at the origin). Indeed, one can quickly check that this torus is given implicitly by the level set . Thus, if we can show that (where ) for any with then we may conclude from the previous theorem. To do this we calculate that
Clearly then we see that if and only if but and since we assumed that we see that . Thus, we see that for every point such that one has that and so the torus is a surface by the previous theorem.
Clearly we know then that an analogous torus centered at any point is a surface, being the image of an example of the tori discussed above, by an affine transformation.
Remark: It’s a curiosity that when we don’t get a surface. In particular, we see that it doesn’t satisfy the ‘smooth’ condition (it’s clearly still a topological surface). There is a very visually obvious reason for this. If one graphs tori with then one gets what is called a ‘horned torus’. This is a torus which doesn’t have a hole, and which curves up ‘inside itself’, ending in a sharp corner. A good picture of this can be found here.
 Carmo, Manfredo Perdigão Do. Differential Geometry of Curves and Surfaces. Upper Saddle River, NJ: Prentice-Hall, 1976. Print.
 Pressley, Andrew. Elementary Differential Geometry. London: Springer, 2001. Print.
 Montiel, Sebastián, A. Ros, and Donald G. Babbitt. Curves and Surfaces. Providence, RI: American Mathematical Society, 2009. Print.