Totally real submanifold of 2 dimensional complex manifold Let $X$  be a 2-dimensional complex  manifold and let $Y$ be a smooth submanifold of $X$ of two real dimension. Let $J$ be the complex structure on the tangent space $T_xX$. The submanifold $Y$ is called totally real if $T_xY\cap J(T_xX)=\{0\}$ for all $x\in Y$.
Among the following submanifolds $Y$: $S^2$, $S^1\times S^1$ and $\mathbb RP^2$, which one can be embedded in $X$ as a totally real submanifold. Why $\mathbb RP^2$ can't be one of them? 

It depends on $X$, but all $3$ of those surfaces can be embedded as totally real submanifolds in some complex manifold of dimension $2$.
In all cases we can use the fact that for a lagrangian submanifold in a Kähler manifold is totally real (which is proven in the answer to this question What is the relation between totally real submanifold and Lagrangian submanifold?).


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*$S^{1} \times S^{1}$ is Lagrangian in $\mathbb{CP}^{2}$ since it is a toric manifold, and the pre-image of a regular value of the toric moment map is a lagrangian torus. 

*$\mathbb{RP}^{2}$ is a lagrangian in $\mathbb{CP}^{2}$, embedded by just taking points whose homogeneous co-ords are real numbers.

*To see $S^{2}$ as a lagrangain, just take a smooth rational curve with self-intersection $-2$ in a del Pezzo surface, which can be achieved by blowing up a point and then a point on its strict transform.
