Projective Space

Question

I have a question and I hope you can help me. I know that the complex projective space, for example ℂ ³ P, consists of the equivalence classes [z] = Z {α: α ∈ ℂ \ {0} and z ∈ ℂ ⁴}. I have a problem where I have the equivalence classes [z] = Z {α: I α ∈ ℝ \ {0} and z ∈ ℂ ⁴}; that is, the equivalence classes are formed by real scalar product and no longer by complex escalar product. I am interested in conserve the complexity of the space and I would like even more being able to relate it to the complex projective space, but it seems I can't do that ... or so I think. I do not know if you could give me your opinion. Thank you.

Answer 1

Your first space is $\Bbb{CP}^3$, a real smooth manifold of dimension 6 (and simultaneously a complex manifold of dimension 3).

Because (as real vector spaces) $\Bbb C^4 \cong \Bbb R^8$, your second space is $\Bbb{RP}^7$, a smooth manifold of dimension 7. Since you're interested in relating this to complex projective space, here are some facts that show their differences:

Real projective spaces of even dimension are non-orientable. Because complex manifolds are of even degree and are canonically orientable (as real manifolds), no real projective space can also be considered as a complex manifold.

Real projective spaces have torsion in their homology groups of even dimension. The homology of complex projective spaces are torsion-free.

I would be surprised if one could embed $\Bbb{CP}^3$ into $\Bbb{RP}^7$, but I can't prove this. I would be interested in a proof of this (or a construction of such an embedding). Similarly for an immersion $\Bbb{CP}^3 \rightarrow \Bbb{RP}^7$.