On canonical embeddings of complex projective spaces in real projective spaces

In this paper, we construct a natural embedding $$\sigma :\mathbb{C}P_\mathbb{R}^{n} \to \mathbb{R}P^{n^2 + 2n} $$ of the complex projective space ℂP n considered as a 2n-dimensional, real-analytic manifold in the real projective space $$\mathbb{R}P^{n^2 + 2n} $$ . The image of the embedding σ is called the ℂP n-surface. To construct the embedding, we consider two equivalent approaches. The first approach is based on properties of holomorphic bivectors in the realification of a complex vector space. This approach allows one to prove that a ℂP-surface is a flat section of a Grassman manifold. In the second approach, we use the adjoint representation of the Lie group U(n + 1) and the canonical decomposition of the Lie algebra u(n). This approach allows one to state a gemetric characterization of the canonical decomposition of the Lie algebra u(n). Moreover, we study properties of the embedding constructed. We prove that this embedding determines the canonical Kähler structure on ℂP ℝ n . In particular, the Fubini-Study metric is exactly the first fundamental form of the embedding and the complex structure on ℂP ℝ n is completely defined by its second fundamental form; therefore, this embedding is said to be canonical. Moreover, we describe invariant and anti-invariant completely geodesic submanifolds of the complex projective space.