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Let's start with a product manifold $M\times N$, with a product Riemannian metric $g_M\oplus g_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and don't depend on the metric. But its gradient vector field does depend on the metric.

Now we would like to change the metric to be $g_\epsilon:= g_M\oplus \epsilon g_N$, with $0<\epsilon< \infty$, and consider the process $\epsilon\to 0$. \infty$. Intuitively, the$N$-component of the gradient will be very small. Consider two critical points$(p_i, q_i)$,$i=1, 2$and the moduli space${\mathcal M}_\epsilon$of trajectories between$(p_i,q_i)$for the metric$g_\epsilon$. So it seems that objects in the limit moduli would be the union of a bunch of trajectories on$M$, of functions$f(\cdot, y_k)$, with several different$y_k$'s. Is that necessary that those trajectories(in$M$) connect successively? Are there other types of limit objects? More precisely, suppose$\epsilon_j\to 0$, \infty$, and $\gamma_j\in {\mathcal M}_{\epsilon_j}$. What are all the possible limits(in a proper sense) of subsequences of $\gamma_j$? If we know all the limits, can we expect a gluing argument to add a good end of the universal moduli $\cup_\epsilon {\mathcal M}_\epsilon$ on the $\epsilon=0$ \epsilon=\infty $side? We may assume that the function and the metrics$g_M$,$g_N$are all generic. This question can be asked more generally: if we have a distribution on a manifold(or a foliation), and we shrink the metric on the direction perpendicular to the foliation, what do the limit Morse trajectories look like? 1 # Morse theory and adiabatic limits Let's start with a product manifold$M\times N$, with a product Riemannian metric$g_M\oplus g_N$. Consider a generic Morse function$f(x, y)$on$M\times N$. Then its critical points are discrete and don't depend on the metric. But its gradient vector field does depend on the metric. Now we would like to change the metric to be$g_\epsilon:= g_M\oplus \epsilon g_N$, with$0<\epsilon< \infty$, and consider the process$\epsilon\to 0$. Intuitively, the$N$-component of the gradient will be very small. Consider two critical points$(p_i, q_i)$,$i=1, 2$and the moduli space${\mathcal M}_\epsilon$of trajectories between$(p_i,q_i)$for the metric$g_\epsilon$. So it seems that objects in the limit moduli would be the union of a bunch of trajectories on$M$, of functions$f(\cdot, y_k)$, with several different$y_k$'s. Is that necessary that those trajectories(in$M$) connect successively? Are there other types of limit objects? More precisely, suppose$\epsilon_j\to 0$, and$\gamma_j\in {\mathcal M}_{\epsilon_j}$. What are all the possible limits(in a proper sense) of subsequences of$\gamma_j$? If we know all the limits, can we expect a gluing argument to add a good end of the universal moduli$\cup_\epsilon {\mathcal M}_\epsilon$on the$\epsilon=0$side? We may assume that the function and the metrics$g_M$,$g_N\$ are all generic.

This question can be asked more generally: if we have a distribution on a manifold(or a foliation), and we shrink the metric on the direction perpendicular to the foliation, what do the limit Morse trajectories look like?