# Does a Riemannian submersion maps horizontal geodesics to geodesics, and a relevant question?

Setup:

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. 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.

UPDATE: I just asked the question on MO, as I haven't received a response yet. If requested, I can certainly delete one of my questions.

Yes, the horizontal lifts of $$N$$-geodesics are $$M$$-geodesics if $$\pi: M\to N$$ is a Riemannian submersion. And yes, a Riemannian submersion commutes with the geodesic flow in horizontal directions.
• Thanks for your answers, but my question 1) was slightly different, it was asking for a proof that projection of horizontal $M$-geodesics are $N$-geodesics. If the proof is long, can you at least outline it or cite a resource that prove it? I'd like to see the proof technique. Commented Mar 19 at 9:01
• I will try to think of a conceptual proof. In the meantime, you can certainly consult a textbook for such basic facts. @LearningMath Your question 1 is not different, because by uniqueness of the geodesic with given initial conditions, a horizonal $M$-geodesic is exactly the same thing as a horizontal lift of an $N$-geodesic. Commented Mar 19 at 9:26