Charts for level set manifolds & Multiplication map $F(A,B)=AB$ from $O(n)\times O(n)\to O(n)$ is smooth

Question

This is homework so no answers please

We have Multiplication map $F:O(n)\times O(n)\to O(n)$ defined as $F(A,B)=AB$ $F:O(n)\times O(n)\to O(n)$, where $O(n)=\{A\in M(n\times n):AA^{t}=id\}$. The smooth structure of O(n) comes from ImpFT: consider smooth map $f:M(n\times n)\to Sym(n\times n)$ defined as $f(A)=A^{t}A$ then $O(n)=f^{-1}(Id)$, where Id is a regular value.

The problem is to show that F is smooth.

Smoothness of F, means smoothness of the coordinate representation of F. So I am trying to unearth the coordinate charts of O(n) from IFT. Any suggestions on how to do that? Any links on doing that for general level-set manifolds.

Thanks

Answer 1

$GL_n$ is a Lie group, in particular multiplication is smooth. $O(n)$ is just a sub manifold which is closed under multiplication, hence $F$ is smooth.

So to prove your homework it would be easier to prove that multiplaction on $GL_n$ is smooth, or even easier (almost trivial) that multiplication on $M(n\times n)$ is smooth. Note that there are no inverses in the latter, hence no group structure. To show that $M(n\times n)$ has a smooth multiplication map, note that the charts are very easy! Write down the function as $R^n \times R^n \to M \times M \to M \to R^n$. You will see smothness.

Answer 2

Two basic theorems in differential geometry would be useful here. Suppose $M,N$ are smooth manifolds and $f\colon M\to N$ is a smooth map.

1. Restricting the domain: If $S\subseteq M$ is an (immersed or embedded) smooth submanifold, then $f|_M\colon S\to N$ is smooth.
2. Restricting the codomain: If $S\subseteq N$ is an embedded smooth submanifold containing $f(M)$, then $f$ is smooth when considered as a map from $M$ to $S$.

Together with the fact that multiplication in $GL(n,\mathbb R)$ is smooth because it's given by polynomials in standard coordinates, this leads to an easy proof.