Is there a result about the action of lie groups in compact manifolds that makes this action transitive?
For example:
$SU(2)$ acts transitively on $CP^1$
$Sp(2)$ acts transitively on $S^4$
$SO(3)$ acts transitively on $S^2$
This action given by left action. I can prove the first example using the following: Consider $v \in CP^1$ and $A \in SU(2)$, the left action give me that $Av= x$, where x is linear combination of the first column of A, as $A$ belong $SU(2)$ its columns are orthonormals and belong $CP^1$. But I would like to find some results about action of Lie group over compact manifolds
Here is one general result you can use to prove transitivity. Suppose that $G$ is a compact Lie group acting on a connectd manifold $X$ and you know the stabilizer $G_x$ of some point $x\in X$. Suppose that you also know that $dim G = dim(X) + dim(G_x)$. Then $G$ acts transitively on $X$. To prove this, observe that $G_x$ is closed in $G$, hence, is a Lie subgroup of $G$ (Cartan's theorem), hence, $G/G_x$ is a manifold of dimension equal to the dimension of $X$. The slice theorem implies that the orbit map $o_x: G\to X, o_x(g)=gx$ descends to an open map $f: G/G_x\to X$. Hence, the image of this map is open in $X$; but $G$ is compact, hence, the image is also closed. Since $X$ is connected, you conclude that the map $f$ is surjective. Hence, the orbit map $o_x$ is also surjective, hence, $G$ acts transitively.