Morse--Smale transversality and surjectivity of the linearized flow operator

Differentiating $\dot{\gamma }+{\mathrm{grad}}_{g}f(\gamma )=0$ in a decaying variation gives the displayed operator by [F1]. Since $\gamma$ converges to the critical points and $\mathrm{\nabla }{\mathrm{grad}}_{g}f$ is continuous, its coefficient along $\gamma$ is bounded; hence $D_{\gamma }:E_{\gamma }\to F_{\gamma }$ is bounded.

By [L2], $\gamma (t)$ approaches $p$ and $q$ exponentially at the two ends. Parallel transport a $g$-orthonormal frame along $\gamma$. In fixed endpoint coordinates its transport equation has coefficients bounded by a constant times $\mathrm{\mid }\dot{\gamma }(t)\mathrm{\mid }$, which are integrable on both tails; hence the frame converges at each end to an orthonormal frame. In this frame, ${\mathrm{\nabla }}_t$ is ordinary differentiation, so $D_{\gamma }$ becomes $z^{\mathrm{\prime }}-A(t)z$, with $A(t)$ converging to $-g_p^{-1}{\mathrm{Hess}}_{p}f$ at $-\mathrm{\infty }$ and $-g_q^{-1}{\mathrm{Hess}}_{q}f$ at $+\mathrm{\infty }$. These limits are self-adjoint for the endpoint inner products and invertible because $f$ is Morse. Parallel transport preserves norms and takes ${\mathrm{\nabla }}_t\xi$ to $z^{\mathrm{\prime }}$, so it identifies exactly the displayed $C_0^1$ and $C_0^0$ spaces, not a weighted Sobolev replacement.

Apply [L1] to step 2.1. Its negative-end decaying initial-value space has dimension $\lambda (p)$, and its positive-end space has dimension $\dim M-\lambda (q)$; therefore $D_{\gamma }$ is Fredholm of index $\lambda (p)-\lambda (q)$. By [L2], differentiating the flow along $W^u(p)$ gives decaying negative-end variational solutions, and differentiating along $W^s(q)$ gives decaying positive-end ones. Both tangent spaces have exactly the corresponding dimensions, so inclusion implies equality with the two ODE initial-value spaces at $\gamma (0)$. The cokernel is consequently isomorphic, through the chosen half-line right inverses, to $T_{\gamma (0)}M\mathrm{/}(T_{\gamma (0)}{W}^u(p)+T_{\gamma (0)}{W}^s(q))$.

The quotient in step 3.1 vanishes exactly when the two tangent spaces span $T_{\gamma (0)}M$, which is transversality. By [L1], that quotient vanishes exactly when $D_{\gamma }$ is onto. This proves the biconditional and the index assertion.