I asked [this question][1] on MSE, but I didn't receive a response yet, so I'm asking here. Apologies if the question is not exactly a research level question, but I'm having some trouble in figuring them out myself. Hints or solutions would be appreciated, as well as resources (book chapter etc.). I did an internet search on this one, but didn't see any affirmative answer.

Let $\pi:(M,g)\to (N,h)$ be a surjective *Riemannian* submersion, 

* i.e. $\forall p\in M, D\pi_p$ is surjective between the respective tangent spaces and that . 
* $T_pM=H_pM \oplus V_pM$ ($g_p$-orthogonal direct sum), and 
* $H_pM$  is isometric to $T_qN, q:=\pi(p)$ via $D\pi_p, i.e. h_q(D\pi_p(v), D\pi_p(w)=g_p(v,w)\forall v,w \in H_pM.$ (this defines the Riemannian submersion part, isometry of the horizontal part of the tangent space with the tangent space of the image/quotient).

**Questions:**

1) **Is it true that $\pi$ maps *horizontal* geodesics in [$M$ to geodesics in $N?$** Perhaps relevant is this MO question: initially horizontal geodesics are always horizontal][1]. I can't help thinking the way we show that an isometry $\phi$ maps geodesics to geodesics: the idea is to show for vectore fields $X,Y$ that $\phi_{*}(\nabla^M_X {Y})= \nabla^N_{\phi_{*}X}{\phi_{*}Y}.$ (John Lee: Riemannian manifolds: an introduction to curvature, P.71). **Should we show the same here for $X, Y$ hotizontal vector fields?**

2) **Is it true that $\forall v \in H_pM, exp^N_q(D\pi_p(v))= \pi(exp_p^M(v)),$ where $exp^M, exp^N$ represent the corresponding exponential maps?** I think this can be proven if 1) is proved, and noting that the initial velocity of the geodesic (if 1) is proven) $t\mapsto \pi(exp_p^M(tv))$ is indeed $D\pi_{p}(v),$ and then using the uniqueness of geodesic with initial point and initial velocity.


  [1]: https://mathoverflow.net/questions/435401/initially-horizontal-geodesic-is-always-horizontal?_gl=1*zy3mgq*_ga*MTg0MDcxMjM1NS4xNjgzOTI2NzMy*_ga_S812YQPLT2*MTcxMDgzNzQ0OS4xOTUuMS4xNzEwODQzMjEwLjAuMC4w