algebraic-topologycomplex-geometrydifferential-geometrymanifolds
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?
The key is to find a set that is the image of two different injective immersions that induce different topologies on it. Here's a hint:
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EDIT: For an example where the two submanifolds are not homeomorphic, try this:
In one case, the submanifold has two connected components, neither of which is compact. In the other, there are three components, one of which is compact.
If your interpretation of "smooth submanifold" is smooth embedded submanifold (meaning that it has the subspace topology), then these two definitions are equivalent. The proof of the existence of a holomorphic slice works just like the smooth case, but using the holomorphic version of the inverse function theorem instead of the smooth one. Here's a sketch of the proof.
Suppose $N\subseteq M$ is a smooth embedded submanifold with a complex structure, and let $\iota\colon N\to M$ denote the inclusion map. Given $p\in N$, we can choose holomorphic coordinates on a neighborhood $V$ of $p$ in $N$, and holomorphic coordinates on a neighborhood $W$ of $\iota(p)$ in $M$. Since $N$ has the subspace topology, after shrinking $V$ and $W$ if necessary we can assume that $V=W\cap \iota(N)$. Since the question is local, we might as well replace $M$ and $N$ by $V$ and $W$, identified with open subsets of $\mathbb C^r$ and $\mathbb C^n$, respectively. Then $d\iota_p$ is an injective complex-linear map from $\mathbb C^r$ to $\mathbb C^n$. After a complex-linear change of coordinates, we may assume that the image of $d\iota_p$ is the span of the first $r$ standard coordinate vectors. Let $Y$ be the span of the complementary $n-r$ coordinate vectors, and define $\Phi\colon \mathbb C^n = \mathbb C^r\times Y \to \mathbb C^n$ by $\Phi(x,y) =\iota(x)+y$. The holomorphic inverse function theorem shows that $\Phi$ is a biholomorphism onto some neighborhood $U$ of $(p,0)$, and $\Phi^{-1}$ is the required local holomorphic chart.
Best Answer
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?).
$S^{1} \times S^{1}$ 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.