Suppose that $X$ and $Y$ are manifolds, with a transitive $G$ action, where $G$ is some Lie group. Suppose that $\phi : X \to Y$ is a surjective submersion, which is also $G$ equivariant.
Is $\phi$ a fiber bundle?
Maybe there are some assumptions that make this true? $\phi$ is not proper. I can assume it's fibers are connected (in fact, contractible).
It is a fibre bundle. You can write $X=G/H$ where $H$ is the stabilizer of an element $x$, and $p_H:G\rightarrow G/H$ is a fiber bundle. You also have $p_L:G\rightarrow G/L=Y$ (where $L$ is the stabilizer of $y$, and $H\subset L$ if you choose $\phi(x)=y$) a fibre bundle. Let $(U_i)$ be a trivialization of $p_L$, $p_L^{-1}(U_i)=U_i\cap L$, $H$ acts on $(p_L^{-1}(U_i))^{-1}$ and the quotient is $U_i\times L/H$ which is isomorphic to $p_H^{-1}(U_i)$.
Fibre bundles for homogeneous spaces