Let $X,Y$ be Riemannian manifolds, and $f\colon X\to Y$ be a Riemannian submersion. Let $\gamma$ be a geodesic on $X$ starting at a point $x\in X$ and which is orthogonal to the fiber $f^{-1}(f(x))$.

**Questions.** (1) Is it true that $\gamma$ is orthogonal to any fiber of $f$ which it intersects?

(2) If this is not true in general, is it true under the extra assumption that $f$ has totally geodesic fibers?

**Remark.** I think this is true in the special case when $X=G$ is a compact Lie group with a bi-invariant metric, and $Y=G/H$ is its homogeneous space with the only metric such that the canonical map $f\colon G\to G/H$ is a Riemannian submersion. (In this case $f$ does have totally geodesic fibers.)

Riemanniansubmersion is not so straightforward/easy to perturb. $\endgroup$ – Alex Degtyarev Jan 25 '15 at 13:16