I need to show that $A = \{(x,s,t) \in \mathbb{R}^3 \; | \; x^4 + s^2 + t^2 = 1 \}$ is diffeomorphic to $B= \{(x, y,s,t) \in \mathbb{R}^4 \; | \; y=-x^2 \text{ and } x^2 + y^2 + y + s^2 + t^2 = 1 \}$.
A map $F: A \rightarrow \mathbb{R}^4$ defined as
$$
F(x,s,t) = (x, -x^2, s, t)
$$
is smooth, injective, and onto $B$.
How do I show that both sets are diffeomorphic? The map $F$ is an immersion, but it's not clear to me how to use this to show that $A$ and $B$ are diffeomorphic.
Best Answer
$A$ is compact, thus the map $f$ which is continuous and injective establishes a homeomorphism from $A$ to $B = f(A)$. You know that $A$ is submanifold of $\mathbb R^3$ and that $f$ is an injective immersion and a homeomorphism onto its image. This implies that $f$ is a diffeomorphism onto its image, i.e. an embedding of smooth manifolds.