Connectedness of fibers of almost Riemannian submersions

Question

EDIT: Let $M,N$ be compact connected smooth Riemannian manifolds. Let us assume that $N$ is closed, while $M$ might have a geodesically convex boundary.

Given $f\colon M\to N$ be an $\varepsilon$-almost Riemannian submersion, i.e. a submersion such that for any point $x\in M$ and any tangent vector $v\in T_xM$ which is orthogonal to the tangent space $T_xF$ to the fiber $F=f^{-1}(f(x))$ one has $$(1-\varepsilon)|v|\leq |df_x(v)|\leq (1+\varepsilon)|v|.$$

Assume in addition that the restriction $f|_{\partial M}\colon \partial M\to N$ is an $\varepsilon$-almost Riemannian submersion.

Let us denote by $D$ the maximal diameter of the fibers of $f$.

Question. Is it true that if $0\leq \varepsilon < 1/100$ and $D< inj(N)/100$ then the fibers of $f$ are connected?

Remark. I think I can prove this in two cases: (1) for $\varepsilon =0$, (2) for closed $M$ and small positive $\varepsilon$.

Answer 1

In the definition of submersion you should assume that $d_xf\colon \textrm{T}_x\to \textrm{T}_{f(x)}$ is onto. In this case there is an almost horizontal lift $\phi_x\colon\textrm{T}_{f(x)}\to\textrm{T}_x$ such that $d_xf\circ\phi_x=\mathrm{id}$.

Moreover, by applying the partition of unity, you can assume that $\phi$ depends continuously on $x$. This makes it possible to lift homotopy from $N$ to $M$.

Since $\textrm{diam}\, F_x$ is small, we can connect any two of its points (say $x$ and $y$) by a short curve. Project this curve to $N$; it is a loop based at $f(x)$. Then shrink the loop to $f(x)$. By lifting this homotopy to $M$, we get a curve from $x$ to $y$ that lies in the fiber.

(This argument is nearly the same as here)