It is well-known that for any Riemannian manifold $M$, the spaces $$H_x(M) = \Bigl\{ \gamma \in AC\big([0, 1], M\big) ~\Big|~ \gamma(0) = x, \int_0^T|\dot{\gamma}(s)|^2 \mathrm{d}s < \infty\Bigr\}$$ and $$H_{xy}(M) = \Bigl\{ \gamma \in H_x(M) ~\Big|~ \gamma(1) = y\Bigr\} $$ of absolutely continuous paths with finite energy (paths of Sobolev regularity 1) have naturally the structure of an infinite-dimensional manifold modelled on a Hilbert space, and the latter is a submanifold of codimension $\mathrm{dim}(M)$, because the end-point-evaluation map given by $\delta(\gamma) = \gamma(1)$ is a surjective submersion from $H_x(M)$ onto $M$.
Now let $\tau$ with $0 = \tau_0 < \tau_1 < \dots < \tau_N = 1$ be a partition of the interval $[0, 1]$ and consider the subsets $$H_{x(y)}^\tau(M) = \{ \gamma \in H_{x(y)}(M) \mid \gamma|_{(t_{j-1}, \tau_j)}~\text{is a geodesic} \}$$ of $H_x(M)$ respectively $H_{xy}(M)$ of piecewise geodesics. It is easy to show that $H_x^\tau(M)$ is a finite-dimensional submanifold of $H_x(M)$.
However, it is not clear if $H_{xy}^\tau(M)$ is a finite-dimensional submanifold of $H_{xy}(M)$: It is not clear if the endpoint evaluation is still surjective when restricted to $H_x^\tau(M)$, because of the problem with conjugate points.
Question: Is $H_{xy}^\tau(M)$ is a finite-dimensional submanifold of $H_{xy}(M)$?
