Understanding smooth manifolds

Question

I'm having a little trouble understanding smooth manifolds.

I have been told the ellipsode $E = x^2+2y^{2}+3z^{2}=6$ Where $(x,y,z) \in\mathbb{R}^3$ is a 2-dimensional manifold of smoothness but I didn't really understand why. Can somebody help explain why? Thanks

Answer 1

You can cover $E$ with an atlas of 2 charts. One is $E - \{(0,0,\sqrt{2})\}$, and the other is $E - \{(0,0,-\sqrt{2}\}$. For both of these charts, the associated maps are a stereographic projection which is a diffeomorphism to a plane.

This is, of course, modulo a lot of details. You should check that the transition map is smooth.

Also, disclaimer: I work with manifolds very little so some of my phrasing and terminology might be a bit off.

Answer 2

You could also use the implicit function theorem to show that this is a manifold. We can consider this ellipsoid as the level set $F^{-1}(0)$ of $F(x,y,z) = x^2 + 2y^2 + 3z^2 - 6$. Note that $DF = [2x$ $4y$ $6z]$. The only point where $DF$ doesn't have maximal rank is at (0,0,0), which doesn't lie on the manifold. So, according to the implicit function theorem, there is always one variable that can be expressed smoothly in terms of the other two i.e. the surface is locally the graph of a smooth function. So it is a smooth manifold.

Answer 3

You don't have to know what a manifold is in order to check that a subset of $\mathbb R^n$ is a submanifold!
To see that $E$ is a submanifold it suffices to check that the smooth function $f(P)=f(x,y,z)= x^2+2y^{2}+3z^{2}-6$ has a non-zero gradient at all $P\in E$, which is clear since $\operatorname {grad} f(x,y,z)=(2x,4y,6z)$ only vanishes at the origin.

An important but neglected distinction
a) Certain subsets $M\subset \mathbb R^n$ are called submanifolds of $\mathbb R^n$: there are several equivalent definitions for these sets, all deriving from the implicit function theorem.
The point I want to emphasize is that (in principle!) given $M$ one can answer the question "is $M$ a submanifold of $\mathbb R^n$ ?" unequivocally with "yes!" or "no!", without invoking any extraneous structure.

b) On the other hand, given an abstract set $S$ it does not make sense to ask whether $S$ is or is not a smooth manifold: a smooth manifold is a structure consisting of of a topology on $S$ plus a complicated set $\mathcal A$, called an atlas, which has to satisfy a lot of weird requirements.

c) The link between the two concepts above is that a submanifold $M\subset \mathbb R^n$ can be endowed very canonically with such an atlas $\mathcal A$, so that $(M,\mathcal A)$ becomes a manifold in its own right.
The important practical point however is that a student asked to check whether some $M\subset R^n$ is a manifold should never worry about atlases or charts: submanifolds are automatically provided with a canonical atlas and the student should only concentrate on the tools for proving that $M$ is a submanifold, like implicit function theorem, submersions, gradients, differentials, immersions, diffeomorphisms,... (these depend of course on the presentation the teacher has chosen).