## Surfaces (Pt. III)

**Point of Post: **This is a continuation of this post.

Now, one of most fascinating, perhaps non-obvious facts about surfaces is that while every subset of which looks locally like the graph of a smooth function is is a surface, the converse is, in a sense, also true. In particular, given any surface and any point on the surface, we may find a neighborhood of which is just the graph of a smooth function . The first step in this is the following which, after unraveled basically says that every point on a surface has a neighborhood which can be paramaterized by a chart such that the inverse of the chart is nothing more than a projection.

**Theorem: ***Let be a surface, , and a differentiable chart at . If then there exists a neighborhood of in and an orthogonal projection such that is open in and is a diffeomorphism.*

**Proof: **Letting denote the coordinate functions of we have that the Jacobian of at is

Now, by assumption that is injective we know that it has a minor which is non-zero, we may assume that the minor is created by the first two columns. Obviously, this assumption can be made since otherwise we may just apply a linear isomorphism of to our map making this conclusion so, but which will not change the conclusion. Consider then the orthogonal projection where we’ve identified with the plane. We see then that is smooth, and by the chain rule

and so by construction we have that . The rest is just a literal statement of the inverse function theorem.

So, now we may draw the conclusion we really want. Namely, think about it. So, since we are able to create a diffeomorphism with open, and we obviously have that the map is a differentiable chart at . But! Notice that we then have that . What does this mean? This means that if we write as coordinate functions for , then

In other words, act trivially on the elements of ! Thus,

Thus, we see that is a neighborhood of in which is just the graph of the smooth function . Noting that we really can’t conclude, a priori that the ‘function part’ of the graph is in the third coordinate, we just didn’t need it for the problem we may conclude the following:

**Corollary: ***Let be a surface. Then, for each there exists an open subset containing , an open subset , and a smooth function such that is the ‘graph’ of .*

*Remark: *The scare quotes around graph, are to mean that all the elements of are of one of the following forms (uniformly, they don’t switch): , , , etc.

As a last note in this post, we’d like to prove that the transition maps between two differentiable charts and on a surface are diffeomorphisms.

**Theorem: ***Let be a surface and let and be two differentiable, intersecting charts on . Then, the transition map given by are diffeomorphisms.*

**Proof: **Since is obviously a smooth homeomorphism, and the inverse of is the transition map it suffices to prove that is smooth. But this follows immediately since we can for a small neighborhood around write as for some orthogonal projection . But, and are smooth, and so the conclusion follows.

**References:**

[1] Carmo, Manfredo Perdigão Do. *Differential Geometry of Curves and Surfaces*. Upper Saddle River, NJ: Prentice-Hall, 1976. Print.

[2] Pressley, Andrew. *Elementary Differential Geometry*. London: Springer, 2001. Print.

[3] Montiel, Sebastián, A. Ros, and Donald G. Babbitt. *Curves and Surfaces*. Providence, RI: American Mathematical Society, 2009. Print.

[…] at . Indeed, we assume for a second that the single-sheeted cone is a surface. We know then from previous theorems that there exists some neighborhood of , some open subset , and a smooth function such that is […]

Pingback by Instructive Non-Examples « Abstract Nonsense | October 14, 2011 |