Let $A\in\mathbb{R}^{n\times n}$ be a matrix. Show that $M=\{v\in\mathbb{R}^n| \langle v,Av\rangle=1\}\subset\mathbb{R}^n$ is a submanifold of dimension $n-1$.
Under which conditions is $M$: a) not empty, b) path connected
I want to show that $M$ is a submanifold. For that I want to use the theorem of the regular value. I struggle a little bit with the scalar product and giving a fitting function
$f:\mathbb{R}^n\to\mathbb{R}$
I tried to make a simple example first:
$n=1$. Then $A\in\mathbb{R}^{1\times 1}$, so $A$ is just an element $x\in\mathbb{R}$. For $v\in\mathbb{R}$ we then have that $\langle v, xv\rangle=xv^2$
Then $f :\mathbb{R}\setminus\{0\}\to\mathbb{R}$, $v\mapsto xv^2$ and $Df(u):\mathbb{R}\setminus\{0\}\to\mathbb{R}$, $v\mapsto 2vu$.
To use the theorem we need to see that $Df(u)$ has to be surjective for every $u$, so we needed to exclude $0$. Also $A\neq 0$ (where I write $0$ for the matrix that has only 0 as entry) is clearly a condition for $M$ beeing not empty.
$Df(u)$ is then obviously surjective for every $u$, and $M=f^{-1}(1)$.
And $M$ is a submanifold of dimension $0$.
Well, in general we should just have:
$f:\mathbb{R}^n\setminus\{0\}\to\mathbb{R}$, $v\mapsto \langle v, Av\rangle$.
How can I write down the jacobian $Df(u)$?
Are my thoughts correct so far? Thanks in advance.
Consider $f(x)=\langle x,A(x)\rangle$ it is the compostion $g\circ h$ where $h:\mathbb{R}^n\rightarrow \mathbb{R}^n\times\mathbb{R}^n$ defined by $h(x)=(x,x)$ and $g:\mathbb{R}^n\times\mathbb{R}^n\rightarrow\mathbb{R}$ defined by $g(u,v)=\langle u,v\rangle$ its derivative at $x$ is $df_x(v)=\langle x,A(u)\rangle+\langle u,A(x)\rangle$.
There exists an open subset $U$ containing $M$ such that the restriction of $f$ to $U$ is strictly positive, (take for example $U=f^{-1}((1/2,3/2))$ If $x\in U$, $df_x(x)=2\langle x,A(x)\rangle\neq 0 $, and $df_x\neq 0$ we conclude that the restriction of $f$ to $U$ is a submersion and $M$ is a submanifold of $\mathbb{R}^n$.