I have a huge problem solving this question. Does anybody have literature on this or maybe a solution?
Let $ M:=\{x\in \mathbb{R}^n : x^{⊤} A x = r\} \subset \mathbb{R}^n $ for $r>0$ and $A$ being a symmetric, positive semidefinite Matrix $A\in \mathbb{R}^{n \times n}$
Is $M$ a $C^l$-Submanifold of $\mathbb{R}^n$. If so, of which Dimension?
There exists a symmetric positive semidefinite matrix $B$ such that $B^2=A$. Let $K$ denote the set of vectors $x$ such that $Bx=0$. Let $L$ be its orthogonal: $B$ induces an isomorphism on $L$.
Then the linear isomorphism $\mathbb{R}^n \rightarrow K \times L$ maps $M$ to $K \times B^{-1}\mathcal{S}_L(r)$. Thus $M$ is a submanifold of dimension $n-1$.