Let $\pi \colon M \to B$ be a riemannian submersion and $g$ the metric on $M$. Then we get the vertical subspace $ \mathcal{V}_x = \ker d_x \pi$ and the horizontal subspace $\mathcal{H}_x= \mathcal{V}_x^\perp$ as the orthogonal space of $\mathcal{V}_x$ in $T_xM$. Now these subspace form the vertical and horizontal bundle and $TM = \mathcal{V} \oplus \mathcal{H}$.
Now considering the map $\pi_* \colon \mathcal{H} \to TB$ as a smooth map between manifolds, is it possible to conclude, that $\pi_*$ is a submersion? Is it a submersion?
More precicsely, I'm having a Lie group $G$ acting properly, freely and by isometries on $M$ and $\pi \colon M \to B =M/G$ is just the quotient manifold, endowed with the submersion metric. I showed that $\pi_* \colon \mathcal{H}/G \to TB$ is a smooth map, which is bijective. But I want to show that it is a diffeomorphisms, Therefore it is necessary and sufficient that $\pi_*$ is a submersion or immersion. But I don't know how to prove this fact.