Q1: Let $f: \mathbb{R}^2 \rightarrow \mathbb{R}$ be given by $f(x,y) = x^2 + y^2 - 1$. Is $f^{-1}[{ 0 } ]$ a submanifold?

A1: Yes, it is by the regular value theorem.

A2: No, 0 is not a regular value.

A3: No, the function is not well-defined.

A4: One needs more information.

Q2: One does one necessarily need for the abstract regular value theorem?

A1: Two smooth manifolds and a smooth map between them.

A2: One smooth manifold and a function on it.

A3: A regular value of a function $f: \mathbb{R} \rightarrow \mathbb{R}$.

Q3: Consider the set of matrices $${ A \in \mathbb{R}^{d \times d} \mid A^T = -A }$$ Is this a manifold?

A1: Yes, it’s of dimension $\frac{d^2 - d}{2}$.

A2: Yes, it’s of dimension $\frac{d(d+1)}{2}$,

A3: No, it is not.