# Initially horizontal geodesic is always horizontal

I am trying to prove the following. (I posted this on math.se with no success)

Let $$E,B$$ be Riemannian manifolds. Suppose $$\pi: E\to B$$ is a Riemannian submersion.

For each $$x\in E$$, define $$V_x E = \ker \pi_{*}$$ and $$H_xE = (V_x E)^{\perp}$$. $$W\in T_x E$$ is said to be horizontal if $$W\in H_x E$$.

Let $$\gamma:[0,1]\to E$$ be a geodesic curve such that $$\gamma'(0)$$ is horizontal. Prove that $$\gamma'(t)$$ is horizontal for all $$t\in [0,1]$$.

My attempt

I mostly followed the proof here(Theorem 3.6).

Firstly, for any smooth curve $$\gamma:[0,1]\to B$$, there is always a locally defined horizontal lift to $$E$$. That is, for any $$p\in \pi^{-1}(\gamma(0))$$, there is $$\epsilon>0$$ and a smooth curve $$\gamma_E:[0,\epsilon) \to E$$ such that $$\pi\circ \gamma_E = \gamma$$, $$\gamma_E(0)=p$$ and $$\gamma_E'$$ is always horizontal.

So, we take a geodesic on $$B$$ with initial condition $$\gamma_B(0)=\pi\circ\gamma(0), \gamma_B'(0)=\pi_{*}\gamma'(0)$$. We can do so (at least locally) through the uniqueness and existence of geodesic.

Then, take a horizontal lift $$\gamma_E:I\to E$$ with $$\gamma_E(0)=\gamma(0)$$. Here, $$I$$ is an open subset of $$[0,1]$$ containing $$0$$.

We can prove that $$\gamma_E$$ is a geodesic. (This argument is not the main topic of this question.)

Then, by the uniqueness of geodesic, we have $$\gamma_E(t) = \gamma(t)$$ for $$t\in I$$.

Here is the question. It seems that we need to prove that $$I$$ is a closed(and therefore clopen) set of $$[0,1]$$, too. Then since $$[0,1]$$ is connected we have $$I=[0,1]$$. But I do not know how to do it.

For example, this book on the proof of Proposition 2.109, (ii),

(Here $$\tilde{c}$$ corresponds to our $$\gamma$$, and $$c$$ corresponds to $$\gamma_B$$)

...Hence the set of parameters where the geodesic $$\tilde{c}$$ is horizontal, and where it is a lift of $$c$$ is an open set containing $$0$$. These two conditions being also closed, they are satisfied on the maximal interval of definition of $$\tilde{c}$$.

Any help?

After choosing local coordinates, by the implicit function theorem (I'm omitting a bunch of technical computations) there is a smooth function $$\varphi:TE \to TE$$ such that $$\varphi(x,-): T_x E \to T_x E$$ is the projection to $$V_x E$$.
Let $$I\subseteq [0,1]$$ be the set on which $$\gamma'$$ is horizontal: this is the set on which $$\varphi\circ \gamma' = 0$$ (the zero section). As a composition of two smooth functions ($$\gamma'$$ is a smooth function from $$[0,1]\to TE$$ and $$\varphi$$ is smooth), the function $$\varphi\circ\gamma'$$ is continuous, and hence $$(\varphi\circ\gamma')^{-1}(0)$$ is closed (the zero section is a closed subset of $$TE$$).