$S^2$ as a totally real submanifold of $\mathbb{CP}^1\times \mathbb{CP}^1$

Question

Can the sphere $S^2$ be embedded in $\mathbb{CP}^1\times \mathbb{CP}^1$ as a totally real submanifold?

Answer 1

The graph of the map $z\to \bar z$, or, more precisely $[z:w]\to [\bar z: \bar w]$, does the job. We check it in the affine part $[z:1]$ . The tangent space $T$ at a point $(z_0, \bar z_0)$ is the set of $(z, \bar z)$, and $i T$ is the set of $(iw, i(\bar w)$ or$(z,-\bar z)$? so $T\cap i T=0$. For the points at infinity, we can use the fact that the map $[z:w]\to [w:z]$ is an holomorphic isomorphism which preserves this graph.