Can every manifold that is locally given as the solution to quadratic equations also be locally represented as the graph of a quadratic function?
If not, is this true for a nice subclass of manifolds? Is there an easy counterexample or a reference for this?
My setting of interest in this is mainly $\mathcal{C}^2$ submanifolds of $\mathbb{R}^n$.
An ellipse is locally represented by solutions to an equations that's quadratic in each of $x$ and $y$, but is nowhere represented as the graph of a quadratic. But maybe that's not what you meant.
A tilted parabola (e.g., $(y+x)^2 = y - x$) is locally represented by a solution to a quadratic (in $x + y$) but is nowhere the graph of a quadratic in either of $x$ or $y$.