Are these two questions asking the same thing?

Question

Question $(I).$

Show that if the $n$-dimensional manifold $M$ is a product of spheres, then there exists an embedding $M \to \mathbb R^{n+1}.$

Question $(2).$

Show that there exists an embedding $S^{n_1} \times \dots \times S^{n_k} \to \mathbb R^{1 + \sum_{i = 1}^{k}n_i}$

Are these two questions asking the same thing? Is $S^{n_1} \times \dots \times S^{n_k}$ necessarily an $n$-dimensional manifold?

Answer 1

Yes.

If $n=n_1+\cdots + n_k$, then $M$ is an $n$-dimensional manifold that is a product of spheres if and only if $M=S^{n_1}\times\cdots\times S^{n_k}$. The two questions are identical.