2

This is from Smooth manifolds and their applications page 7:

Let $P=P^r$ be a subset of the smooth manifold $M^k$ of class $m$, defined near each of its points by a system of $k-r$ independent equations. This means that for each point $a \in P$ there is a nbhood $U =U^k$ in $M=M^k$ with local system $X$ such that the intersection $P \cap U$ consists of all points whose coordinates satisfy the equations $$ \psi^j (x^1, ..., x^k) = 0; j=1,...,k-r$$

I am trying to understand what it means. Can someone help me with the example I am trying to construct? If $M=S^3$ and $P$ is the equator then what would the $\psi^j$ be? Basically, I'd like to understand what this whole section "F)" is about.

snailspace
  • 857

1 Answers1

2

It depends on what you mean by "the equator" on $S^3$.

If you're thinking about the set of all points equidistant from the north and south poles, which is a $2$-dimensional submanifold diffeomorphic to $S^2$, then you can take $\psi^1$ to be (the restriction of) the 3rd coordinate function $x^3$.

If you're thinking of an equatorial great circle (such as the set of points where $x^2 = x^3=0$), then you can take $\psi^1 = x^2$ and $\psi^2 = x^3$.

Jack Lee
  • 46,803