Suppose a smooth manifold $M^n$ can be immersed in $E = \mathbb{R}^k \times \mathbb{S}^1 \times \cdots \times \mathbb{S}^1$ with codimension $1$, where there are, say, $l$ factors $\mathbb{S}^1$.
Is it true that $M$ can also be immersed in $\mathbb{R}^k \times \mathbb{R}^l$ with codimension $1$? I suspect this is the case, for this latter space covers $E$ and one could lift the immersion. Is this correct? Could you help me out here?
Note that $\mathbb{R}\times S^1$ embeds in $\mathbb{R}^2$ via the map $(x, z) \mapsto e^xz$. Repeated application of this map gives an embedding $\varphi : \mathbb{R}^k\times(S^1)^l \to \mathbb{R}^{k+l}$.
If $f : X \to \mathbb{R}^k\times(S^1)^l$ is an immersion, then $\varphi\circ f : X \to \mathbb{R}^{k+l}$ is also an immersion (because $\varphi$ is an embedding). Moreover, the codimension of the immersion $f$ is equal to the codimension of the immersion $\varphi\circ f$ (because $\dim \mathbb{R}^k\times(S^1)^l = \dim \mathbb{R}^{k+l}$).