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).
Define $S:=exp_p(H_pM)\subset M.$
My questions are:
- Is $S$ an embedded totally geodesic submanifold of $M?$ It seems to me yes, since it has been defined via the exponential map at $p.$ But while attempting to prove it, I'm running into a problem: say $p':=exp^M_p(v)\in S$ where $v\in H_pM,$ and then let $w\in T_{p'}S \subset T_{p'}M.$ How do I show that the $S$-geodesic $t\mapsto exp^S_{p'}tw$ is also an $M$ geodesic, i.e. $\exists u\in T_{p'}M$ so that $exp^S_{p'}tw = exp^M_{p'}tu$? An illustrative diagram is below:
Ideas I thought about: It seems $T_{p'}S$ should be isometric to $H_{p'}M,$ but I'm not sure how? Should I use parallel transport along the geodesic joining $p,p'$ to transport $w$ back to a vector in $H_pM?$
- Is $S$ isometric to $N$ via $\pi?$ I'm guessing no, but I'm having problem to see why not? A counter example would be nice.
Resources or solutions would be greatly appreciated!
As already pointed out in the comments, $S$ will generally not be totally geodesic (because in "transverse" directions it is not necessarily horizontal). You should keep some basic examples in mind when thinking of such questions. Thus, if you consider the Hopf fibration $S^3\to S^2$, it should be obvious that the answer to your second question is negative.