I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two submanifolds of $N$, and we consider the corresponding conormal bundles $\nu^{\ast} S_0$ and $\nu^{\ast} S_1$. Let $H\colon T^{\ast} N\times I\to\mathbb{R}$ be a smooth function and $X_H$ the corresponding Hamiltonian vector field, and $\phi_H^t$ the corresponding Hamiltonian flow. We define a function $g\colon\nu^{\ast} S_1\to\mathbb{R}$ by $$ g(x)=\mathcal{A}_H (z_H^x), $$ where $\mathcal{A}_H$ is the corresponding action functional, and $z_H^x$ the path given by $z_H^x (t)=\phi_H^t\circ(\phi_H^1)^{-1}(x)$. Then he claim that:
- The intersection $\phi_H^1 (\nu^{\ast} S_0)\cap\nu^{\ast} S_1$ is compact in $T^{\ast} N$.
It's obviously not the case: if we pick $H=0$, then the condition is equivalent to saying that $S_0$ and $S_1$ are disjoint. But this is a procedure to his proof of the fact that the action spectrum is compact and has measure zero. So I'm wondering if
1'. The image $g(\phi_H^1 (\nu^{\ast} S_0)\cap\nu^{\ast} S_1)$ is compact in $\mathbb{R}$.
is true.
Another claim is about the linearization of the ODO $\frac{\partial }{\partial t} -X_H$, namely $\nabla_{\dot{\gamma }} -DX_H (\gamma)$. We consider the path space with boundary conditions $\Omega_{S_0 S_1} (T^{\ast} N,\mathrm{d}\lambda)$, given by all paths starting from $\nu^{\ast} S_0$ and ends at $\nu^{\ast} S_1$. Then the tangent space of this path space at a path $\gamma$ is given by $\Gamma_{S_0S_1} (\gamma^{\ast} (TT^{\ast} N))$, the space of vector fields on $\gamma$ with endpoints in the given tangent spaces. The linearization is then a linear operator $\nabla_{\dot{\gamma}} -\mathrm{D} X_H\colon\Gamma_{S_0S_1} (\gamma^{\ast} (TT^{\ast}N))\to\Gamma (\gamma^{\ast} (TT^{\ast} N))$. Then the book claims that
- The intersection $\phi_H^1 (\nu^{\ast} S_0)\cap\nu^{\ast} S_1$ is transversal if and only if the linearization $\nabla_{\dot{\gamma }} -\mathrm{D}X_H$ is surjective.
I have no idea why this is true. I don't know how to relate the surjectivity of the linearization to the transversality of intersections. Any comments or answers are helpful.
