## Short notes

8. (Alejandro) Let $f:X\to X$ be an arbitrary transitive homeomorphism and $u:X\to(0,1/4)$ be an arbitrary non-constant continuous function. Then, let’s define $c(x):=u(x)-u(fx)+1$, $x\in X$, and consider the suspension flow $f_t:X_c\to X_c$on $X_c$. Note that for each $x\in X$ and $t\in(0,1/4)$: $f_1(x,t+u(x))=(x,t+u(x)+1)=(x,t+u(fx)+c(x))=(fx,t+u(fx))$. So the compact set $\lbrace(x,t+u(x)):x\in X\rbrace$ is $f_1$-invariant for every $t\in(0,1/4)$, and $f_1$ is not transitive. Notice the function c is not constant because $f$ is transitive and $u$ is not constant itself.

7. Let $A\in\mathrm{GL}(d+1,\mathbb{R})$ and $f:\mathbb{P}^{d}\to \mathbb{P}^{d}$ be the projective action induced by $A$. Then for $\xi=[v]\in \mathbb{P}^{d}$, $u\in T_\xi \mathbb{P}^{d}$, $D_\xi f(u)=\frac{\mathrm{proj}_{Av}(Au)}{\|Av\|}$.

So $\|D_\xi f\|=\sup_{\|u\|=1}\|D_\xi f(u)\|\le \frac{\|A\|}{\|Av\|}$, and $\|Df\|=\sup_{\xi}\|D_\xi f\|\le\frac{\|A\|}{m(A)}$.
Replace $A$ by $A^{-1}$: $\|D_\xi f^{-1}\|\le \frac{\|A^{-1}\|}{\|A^{-1}v\|}$, and $\|Df^{-1}\|=\sup_{\xi}\|D_\xi f^{-1}\|\le\frac{\|A^{-1}\|}{m(A^{-1})}$,
or: $m(D_\xi f)\ge\frac{\|A^{-1}v\|}{\|A^{-1}\|}=\|A^{-1}v\|\cdot m(A)$, and $m(Df)\ge\frac{m(A^{-1})}{\|A^{-1}\|}=\frac{m(A)}{\|A\|}$.
Replace $f$ be $f^n$ and pass $n\to\infty$: $\lambda^-(A)-\lambda^+(A)\le \lambda^-(f,\xi)\le \lambda^+(f,\xi)\le\lambda^+(A)-\lambda^-(A)$.

Let $A\in\mathrm{SL}(d+1,\mathbb{R})$. It seems that $f$ preserves the volume on $\mathbb{P}^{d}$ iff $A\in\mathrm{O}(d+1)$.

6. Let $(M,g)$ be a Riemannian manifold and $\phi_t:TM\to TM$ be the geodesic flow.
Let $\pi:TM\to M$ be the canonical projection and $V(\theta)=\ker(d_\theta\pi)$ be the vertical subspace.
Let $K_\theta:T_\theta TM\to T_{x}M$, $\xi\mapsto (\nabla_{\dot{\gamma}}\theta)(0)$, where $\theta:[-\epsilon,\epsilon]\to TM$ with $\dot{\theta}(0)=\xi$ and $\gamma=\pi\circ\theta$. Let $H(\theta)=\ker(K_\theta)$.

Then $T_\theta TM=V(\theta)\oplus H(\theta)$, and $d_\theta\pi:H(\theta)\to T_{x}M$, $K_\theta:V(\theta)\to T_xM$ give the identifications and induce the following
$\bullet$ (geodesic vector) $G:TM\to TTM, \theta\mapsto (\theta,0)$,
$\bullet$ (Sasaki metric) $\hat{g}_\theta(\xi,\eta)=g_x(\xi_h,\eta_h)+g_x(\xi_v,\eta_v)$,
$\bullet$ (symplectic form) $\omega_\theta(\xi,\eta)=g_x(\xi_h,\eta_v)-g_x(\eta_v,\xi_h)$,
$\bullet$ (almost complex structure) $J:T_\theta TM\to T_{\theta}TM, \xi=(\xi_h,\xi_v)\to (-\xi_v,\xi_h)$,
$\bullet$ (contact form on $SM$) $\alpha_\theta:T_\theta SM\to\mathbb{R},\xi\mapsto \hat{g}_\theta(\xi,G(\theta))=g_x(\xi_h,\theta)$.

5. Let $c:S^1\to\mathbb{T}^2$ be an embedding and $(a,b)$ be the homotopy class of $c$. Then $(a,b)$ is coprime.

4. (Etnyre and Ghrist pdf) Unlike foliations, contact structures are structurally stable, in the sense that not only is a perturbation of a contact form $\alpha$ still a contact form, but also such a perturbation has kernel isotopic to that of $\alpha$. In fact, a standard application of the Moser method in this context implies that every contact structure is locally contactomorphic to, that is, diffeomorphic via a map which carries the contact structure, to the kernel of $dz+xdy$ on $\mathbb{R}^3$.

3. Mitsumatsu’s Theorem (told by YU Bin).
Let $M$ be a 3-manifold supporting some Anosov flow. Then $M$ admits a contact structure.
Proof. Let $f_t:M\to M$ be an Anosov flow and $E^s\oplus\lbrace X\rbrace\oplus E^u$ be the splitting. Let $e^s$ and $e^u$ be two unit vector fields, and $L_x=\lbrace X(x),e^s_x+e^u_x\rbrace$. Then it is easy to check that $L$ is non-integrable at every $x$. So by 1., there exists some contact form.

Since $\alpha(X)=0$, and $d\alpha(X,\cdot)\neq0$, we get
$\frac{d}{dt}{f_t\alpha}=\mathcal{L}_X(\alpha)=d\circ i_X(\alpha)+i_X(d\alpha)=d\alpha(X,\cdot)\neq0$. So the problem is, $f_t$ does not preserve $\alpha$. In fact there exists non-transitive Anosov flow constructed by Franks and Williams (here), which can not preserve any contact form.

2. Let $(M,\alpha)$ be a contact manifold. Then there exists a distinguished vector field $V$, called the characterisitc vector field or Reeb field, which is determined uniquely by the following two conditions: $\alpha(V)=1$, $\imath_V d\alpha=0$. The corresponding flow $\phi_t$ generated by $V$ is called the contact flow.

Proposition. The contact flow always preserves the contact form. That is, given a contact form $\alpha$ on $M$, we have $\phi_t^\ast\alpha=\alpha$. Equivalently, $\alpha(D\phi_t(v))=\alpha(v)$ for all $v\in TM$, $t$.
Proof. $\frac{d}{dt}{\phi_t\alpha}=\mathcal{L}_V(\alpha)=d\circ i_V(\alpha)+i_V(d\alpha)=d(\alpha(V))+d\alpha(V,\cdot)=0$.

On the other hand, let’s start with a flow $f_t$ preserving $\alpha$. We see that $\alpha(X_{f_tx})=\alpha(Df_t(X_x))=\alpha(X_x)$, $\alpha(X)$ is an $f_t$-invariant smooth function. So if the flow $f_t$ is transitive (for example a vol-pre Anosov flow), then $\alpha(X)$ must be constant (up to scaling, we can assume $\alpha(X)=1$) and hence $d\alpha(X,\cdot)=\mathcal{L}_X(\alpha)-d(\alpha(X))=0$, which implies that $f_t=\phi_t$.

1. Let $M$ be a closed 3 dimensional manifold, $\alpha$ be a nonvanishing 1-form on $M$. Then $\alpha$ is said to be a contact form if the following holds:
$\alpha\wedge d\alpha$ is nonvanishing at every $x$.

Let $L:M\to G_2(TM)$ a plane-field. It is said to be a contact structure if it is a kernel of the contact form: $L_x=\ker \alpha_x$.

The orientation induced by $\alpha\wedge d\alpha$ is independent of $\alpha$.
Hence $L$ has a natural orientation, either agree (a positive structure) or disagree (a negative structure) with that of $M$.

Note: a plane-field $L:M\to G_2(TM)$ is a contact structure iff $L$ is non-integrable at every $x$.

Proof. (1) Let $L_x=\ker \alpha_x$ for each $x$. Pick two local fields $X,Y$ on $U$ spanning $L_U$. Extends to a basis $\lbrace X,Y,Z\rbrace$ of $T_UM$. Then
$0\neq\alpha\wedge d\alpha(Z,X,Y)=\alpha(Z)\cdot d\alpha(X,Y)=\alpha(Z)\cdot \alpha([X,Y]).$
So $\alpha([X,Y])\neq 0$ over $U$. This shows that $L$ is non-integrable at every $x$.

(2) Let $L:M\to G_2(TM)$ be a plane-field and $\alpha$ be a one form with $L_x=\ker \alpha_x$ for every $x$ such that $L$ is non-integrable at every $x$. Let $X,Y,Z$ be given as above. Then
$\alpha\wedge d\alpha(Z,X,Y)=\alpha(Z)\cdot d\alpha(X,Y)=\alpha(Z)\cdot \alpha([X,Y])\neq 0,$
since $[X,Y]\notin L$. So $\alpha\wedge d\alpha$ is nonvanishing at every $x$.

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According to the Frobenius integrability theorem, a contact structure is a maximally non-integrable plane-field. In particular, a contact structure is locally twisted at every point and may be thought of as an anti-foliation.