How Can I construct an Embedding for $S^n\times \mathbb{R} \rightarrow \mathbb{R}^{n+1}$

Question

I will like to construct an embedding for $S^n\times \mathbb{R} \rightarrow \mathbb{R}^{n+1}$.

I have constructed embeddings for similar question (e.g. $S^1\times S^1 \rightarrow S^3$), but I can’t seem to construct one for this question. Your help would be greatly appreciated.

Answer 1

The key is to observe that $$h : S^n \times (0,\infty) \to \mathbb R^{n+1} \setminus \{0\}, h(x, t) = t x$$ is a diffeomorphism. Its inverse is $$g : \mathbb R^{n+1} \setminus \{0\} \to S^n \times (0,\infty), g(y) = (\frac{y}{\lVert y \rvert},\lVert y \rvert) .$$

Then $$\psi : S^n \times \mathbb R \to \mathbb R^{n+1}, \psi(x,t) = (\ln t)x$$ is an embedding.

Note that all maps occurring here are smooth.