## Tag Archives: contact form

### Symplectic and contact manifolds

Let $(M,\omega)$ be a symplectic manifold. It said to be exact if $\omega=d\lambda$ for some one-form $\lambda$ on $M$.

(1) If $\omega=d\lambda$ is exact, then there is a canonical isomorphism between the v.f. and 1-forms. In particular, there exists a v.f. $X$ such that $\lambda=i_X\omega$. Then we have $\lambda(X)=\omega(X,X)=0$, and $L_X\lambda=i_X d\lambda+d i_X\lambda=i_X\omega +0=\lambda$, and $L_X\omega=d i_X\omega=d\lambda=\omega$.

(2) Suppose there exists a vector field $X$ on $M$ such that its Lie-derivative $L_X\omega=\omega$ (notice the difference with $L_X\omega=0$). Then Cartan’s formula says that $\omega=i_X d\omega+ di_X\omega=d\lambda$, where $\lambda=i_X\omega$. So $\omega=d\lambda$ is exact, and $L_X\lambda=i_Xd\lambda+di_X\lambda=i_X\omega+0=\lambda$.

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.