I have trouble with these exercises. I hope you can give me advice o how to try them or books where these themes are explained better.
For each of the following sets determine for which values of the parameter $a$, $S_a$ is a smooth surface, its dmension and if is connected: a) $S_a=\{(x,y)\in\mathbb{R}^2:x^2-y^2=a\}$; b)$S_a=\{(x,y,z)\in\mathbb{R}^3:x^2-y^2=a\}$; c)$S_a=\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=z^2+a\}$.
And the same for this one:
How many maps (or letters) is the minimal quantity in order to form an atlas of the following surfaces: a) $\{(x,y,z)\in\mathbb{R}^3: z^2+y^2+z^2=1\}$; b) $\{(x,y,z)\in\mathbb{R}^3:x^2+y^2=1\}$
The definitions of the notes of the teacher are the following:
$S\subseteq\mathbb{R}^n$ is a topological surface of dimension $k$ if every point of $S$ has a neighborhood $U$ in $S$ homeomorphic to an open subset in $\mathbb{R}^k$. $(U,\xi)$ with $U$ an open subset in $S$ and $\xi$ a homeomorphism of $U$ onto an open subset of $\mathbb{R}^k$ is a letter (or map) in $S$. A family $A$ of maps in $S$ is an atlas of $S$ if $\bigcup\{U:(U,\xi)\in A\}=S$; $S\subseteq\mathbb{R}^n$ is a smooth surface of dimension $k$ of class $C^l$ if $S$ is a topological surface of dimnsion $k$ and $S$ has an atlas of class $C^l$.
What I tried for the first exercise was to take a point of the set, which clearly is an hyperbole and try to find an open set conataining the point and a function defined on it with rank in $\mathbb{R}$ such that it was a homeomorphism but I didn't know how to do it.