Q1: Let $S^1$ be the one-dimensional sphere and $i: S^1 \rightarrow \mathbb{R}^2$ the inclusion map. Is $i$ differentiable at $\binom{0}{1}$?

A1: Yes, since we can use the chart $h( \binom{x_1}{x_2} ) = x_1$ for $\binom{x_1}{x_2} \in S^1 \setminus { \binom{0}{-1} } $ and then $i \circ h^{-1}$ is differentiable at $\binom{0}{1}$.

A2: No, there could be charts where $i \circ h^{-1}$ is not differentiable at $\binom{0}{1}$.

A3: One needs more information.

Q2: Let $S^1$ be the one-dimensional sphere and $P^1(\mathbb{R})$ the one-dimensional projective space. Is $k$ given by $k( \binom{x_1}{x_2}) = \frac{x_1}{x_2}$ on a suitable domain a chart for $P^1(\mathbb{R})$.

A1: Yes, it is.

A2: No, it isn’t.

A3: One needs more information.

Q3: Let $S^1$ be the one-dimensional sphere and $P^1(\mathbb{R})$ the one-dimensional projective space and $p: S^1 \rightarrow P^1(\mathbb{R})$ the canonical projection. Let $h$ given by $h( \binom{x_1}{x_2} ) = x_1$ for $\binom{x_1}{x_2} \in S^1 \setminus { \binom{0}{-1} } $ and $k$ given by $k( \binom{x_1}{x_2}) = \frac{x_1}{x_2}$ be a chart for $P^1(\mathbb{R})$. What is $k \circ p \circ h^{-1}$?

A1: $$ x_1^\prime \mapsto \frac{x_1^\prime}{\sqrt{1 - (x_1^\prime)^2}} $$

A2: $$ x_1^\prime \mapsto x_1^\prime $$

A3: $$ x_1^\prime \mapsto (x_1^\prime)^2 $$

A4: $$ x_1^\prime \mapsto \frac{x_1^\prime}{\sqrt{1 + (x_1^\prime)^2}} $$