Explanation of defination of Manifold

Question

While reading the book on Forms and connection ,I am stuck with following defination of manifold.I am stuck at the part after defining function $f$ for submersion. Can anyone explain me this defination using example?

Answer 1

The idea is to see the submanifold as the solution to a system of $k$ equations with $n+k$ unknowns, at least locally. Requiring that $f$ be a submersion is asking that the equations not be redundant.

Consider the example of a circle in $\mathbb{R}^2$: you can think of it as the solutions to the equation $$ x^2 + y^2 = 1. $$ In this example you can take $U$ to be $\mathbb{R}^2 \setminus \{ 0 \}$ and your submersion to be $f \colon U \to \mathbb{R}$ by $f(x,y) = x^2+y^2-1$.

Notice that the differential of $f$ is $2x dx + 2y dy$, which vanishes only at the origin, so this is a submersion.

We can consider a circle inside of $\mathbb{R}^3$ also, for instance, by taking a single set $U = \mathbb{R}^3 \setminus ( \{ (0,0) \} \times \mathbb{R} )$ and taking $ f \colon U \to \mathbb{R}^2$ by $f(x,y,z) = (x^2+y^2-1, z)$. The circle is then the set of points with $z=0$ and $x^2+y^2 = 1$.

Now, the idea of allowing yourself several open sets and several different submersions is that your submanifold might only look like this locally. (If it is globally obtainable by a submersion, it has what is called a trivial normal bundle. If the submanifold is something non-orientable, like a Klein bottle or real projective space sitting inside $\mathbb{R}^4$, you will need multiple open sets and multiple functions. There are other examples, as discussed in this related question: Manifold embedded in euclidean space with nontrivial normal bundle)