The continuation map of constant data is the identity

Given: The Axiom of Choice, a Morse--Smale pair $(f,g)$ on a closed manifold $M$ and its constant continuation datum.

For the constant datum, all continuation solutions are full autonomous negative-gradient curves with the prescribed limits. Their nonconstant subset consists exactly of the Morse trajectories of A Morse trajectory from one critical point to another; constant critical curves are additional solutions (A regular continuation datum between Morse--Smale pairs). The energy identity gives ${\int }_{\mathbb{R}}\mathrm{\mid }\dot{u}{\mathrm{\mid }}^2=f(p)-f(q)$, so equal endpoints force a constant curve (The continuation energy identity).

At nonconstant solutions, regularity is the Morse--Smale transversality condition in the endpoint fibre-product description. At the constant solution $u_p$, the operator is $d\mathrm{/}ds+H_p$, with $H_p$ the invertible self-adjoint Hessian endomorphism; its decaying negative- and positive-end spaces are the complementary negative and positive Hessian subspaces, so it is onto by Fredholm index and range of an asymptotically hyperbolic first-order operator. Thus the constant datum is regular. For a regular datum the moduli space $\mathcal{C}(p,q)$ is a smooth manifold of dimension $\mathrm{ind}(p)-\mathrm{ind}(q)$, empty when this number is negative (A regular continuation datum between Morse--Smale pairs).

Every translate $s\mapsto u(s+c)$ of a solution of the constant datum solves the same autonomous equation with the same limits. A nonconstant solution yields a continuous nonconstant translation family: if the family were constant, evaluating at one fixed time would make $u$ constant. This argument applies to the actual metric equation directly, including curves with coincident endpoint labels.

At a critical point $p$ the constant curve solves the equation. In the fixed-window orientation sequence, the unstable input maps identically to the stable normal quotient under the Hessian splitting; with matched endpoint rays its kernel sign is $+1$ (Orientation lines orient the continuation moduli spaces compatibly with gluing).