Consider the projective space $\mathbb R P^n$, constructed as the quotient space of the sphere $S^n \subseteq \mathbb R^{n+1}$ with the equivalence relation where we identify opposite points.
I now want to prove or disprove that the set $M := \{(x, [y]) : [y] \in \mathbb R P^n, x \in [y]\}$ is a submanifold of the manifold $N := \mathbb R^{n+1} \times \mathbb R P^n$.
Now I rather suspect that it's not a submanifold, but I wasn't able to prove it so far. My thinking was, $M$ is a submanifold of $N$ if and only if there is an injective immersion $\iota: M \to N$. Now the first choice for such an immersion would be the natural map $M \to N, (x, [y]) \mapsto (x, [y])$. To check if that's an immersion, we would need to check that the Jacobean matrix is injective. I have not proven it yet but I suspect it will not be injective, basically because the components of a vector $x$ of the first component of $(x, [y])$ also appear in the $[y]$-component so the Jacobean it shouldn't add up to full rank?
But let's assume that's true, then I would have shown that the identity map $M \to N$ isn't an immersion... how would I go about showing that there doesn't exist any other map $M \to N$ that happens to be an injective immersion? Is there an easy way to see on why this is or isn't the case?
The most similar question I could find was this one and the set in question is not the same there as far as I can tell.
Yes it is a submanifold, since the projection $M \to \Bbb RP^n$ is a line bundle. You can trivialize this vector bundle easily on the usual open set $U_i = \{[y_0: \dots : y_n] \in \Bbb RP^n : y_i \neq 0\}$. This will give you a local parametrization of $M$, and this shows that $M$ is a submanifold of $N$.