I have been trying to solve this problem for a long while now and not getting far.
Problem: Let $n \geq 3$. Characterize all vectors $a, b, c \in \mathbb{R}^n$ such that $$M := \{x \in \mathbb{R}^n : \langle a, x \rangle = 0,\langle b, x \rangle = 0 \ \text{and} \ |\langle c, x \rangle| < 1 \} $$ is a $k$-dimensional submanifold of $\mathbb{R}^n$ for $k \in \{1,...,n-1\}$
Any help/hints, please?
As $\langle a, x\rangle = 0$ and $\langle b, x\rangle = 0$, we see that $x \in \operatorname{span}\{a, b\}^{\perp}$ which is a subspace of $\mathbb{R}^n$ of dimension
$$n - \dim\operatorname{span}\{a, b\} = \begin{cases} n - 2 & a, b\ \text{are linearly independent}\\ n - 1 & a, b\ \text{are linearly dependent but not both zero}\\ n & a = b = 0. \end{cases}$$
Now note that if $x \in \operatorname{span}\{a, b\}^{\perp}$, then $\langle c, x\rangle = \langle c', x\rangle$ where $c'$ is the orthogonal projection of $c'$ onto $\operatorname{span}\{a, b\}^{\perp}$. Therefore
$$\{x \in \mathbb{R}^n : \langle a, x\rangle = 0, \langle b, x\rangle = 0, |\langle c, x\rangle| < 1\} = \{x \in \operatorname{span}\{a, b\}^{\perp} : |\langle c', x\rangle| < 1\}.$$
As $|\langle c', x\rangle| < 1$ is an open condition, we see that $\{x \in \operatorname{span}\{a, b\}^{\perp} : |\langle c', x\rangle| < 1\}$ is an open subset of $\operatorname{span}\{a, b\}^{\perp}$ and hence a submanifold of $\mathbb{R}^n$ of dimension $\dim\operatorname{span}\{a, b\}^{\perp} \in \{n - 2, n - 1, n\}$ depending on $a$ and $b$ as determined above; note, the dimension does not depend on $c$.