Immersion $\mathbb{S}^n\times\mathbb{R}\to\mathbb{R}^{n+1}$

Question

Immersion $\mathbb{S}^2\times\mathbb{R}\to\mathbb{R}^3$

As $\mathbb{S}^2\times\mathbb{R}$ it's not compact, i can give immersion given by $$(x,y,z,t)\to e^t(x, y, z).$$

or i'm wrong. Could you give me some other immersion.

Answer 1

This is too long for a comment, but I can't help adding some more explicit hints to @Ted Shifrin comment.

1) Any manifold $M\subset\mathbb R^m$ has an open tubular nbhd diffeomorphic to the normal bundle $\nu M$ of $M$ in $\mathbb R^m$.

2) If a manifold is defined in some open ngbd $U\subset \mathbb R^m$ by a submersion $U\to\mathbb R^k$ its normal bundle is trivial: the gradients $\nabla f_i$ give a diffeo $M\times\mathbb R^k\equiv\mathbb\nu M\equiv U$.

For a sphere $M=\mathbb S^{p}$ this is the content of the initial question with $U=\mathbb R^{p+1}\setminus\{0\}$.

3) If $M$ is defined by a submersion as in 2), then the product $M\times N$ with any manifold $N\subset\mathbb R^n=\mathbb R^k\times\mathbb R^{n-k}$, $n\ge k$, can be embedded in $\mathbb R^{m+n-k}$.

4) Argue by induction with $Y=\mathbb S^{p_1}\times\cdots\times\mathbb S^{p_r}$ and $X=\mathbb S^{p}$.

This is a quite simple argument for the beautiful fact that any product of spheres is a hypersurface.