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_con 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.

——————————-

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.

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