Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 15 / 4

For a covector $p\in T_x^{\ast }X$, define the cotangent lift of a diffeomorphism byThe inverse derivative here maps $T_{f(x)}X$ to $T_{x}X$. This is a smooth diffeomorphism, its inverse is $(f^{-1}{)}_{\#}$, and it satisfies $\pi \circ f_{\#}=f\circ \pi$.

For $v\in T_{(x,p)}{T}^{\ast }X$, the pullback of a differential form and the defining formula for the Liouville one-form giveHence $f_{\#}^{\ast }\alpha =\alpha$, and applying the exterior derivative also proves $f_{\#}^{\ast }\omega =\omega$. The cotangent lift of a diffeomorphism thus preserves both canonical forms, with no extra choice of metric or connection.

First observe that the Liouville one-form is zero as an ambient covector exactly on the zero section. Indeed, ${\alpha }_{(x,p)}=p\circ d\pi$, and $d\pi$ is surjective. Since $dg$ is invertible, $g^{\ast }\alpha =\alpha$ therefore implies that $g$ maps the zero section onto itself. It induces a diffeomorphism $f:X\to X$ defined by $g(x,0)=(f(x),0)$.

The map $g$ is a symplectomorphism, since it preserves $-d\alpha$. Let $E={\sum }_i{p}_i{\partial }_{p_i}$ be the vertical Liouville vector field. With the chosen sign convention,Preservation of both $\omega$ and $\alpha$ forces $g_{\ast }E=E$ because $\omega$ is nondegenerate. Its complete cotangent fiber-dilation flow isUniqueness of integral curves gives $g\circ D_s=D_s\circ g$. If $g(x,p)=(y,P)$, let $s\to -\infty$. Continuity and this commutation yieldThus $y=f(x)$ for every $p$: the whole cotangent fiber over $x$ maps into the fiber over $f(x)$. This step proves that $g$ covers $f$; it was not assumed.

Now write $g(x,p)=(f(x),P(x,p))$. For any $u\in T_{x}X$, choose a tangent vector to $T^{\ast }X$ projecting to $u$. Evaluating $g^{\ast }\alpha =\alpha$ on it givesSince $d{f}_x$ is invertible, $P(x,p)=p\circ (d{f}_x{)}^{-1}$. Thereforeand $f$ is unique. This rigidity of cotangent Liouville-form preservation uses smoothness at the zero section; it follows from the canonical form and its dilation dynamics on the entire cotangent bundle.

For any positive-dimensional $X$, choose a smooth function $h$ with $dh$ not identically zero. A nonconstant smooth bump in a coordinate chart supplies one on an arbitrary smooth manifold. Translate by this exact differential form:The cotangent fiber translation identities giveThus a nontrivial exact fiber translation is a symplectomorphism that fails to preserve the Liouville one-form. For example, on $T^{\ast }\mathbb{R}$ one can take $T(q,p)=(q,p+1)$: it preserves $dq\wedge dp$ but changes $pdq$ to $(p+1)dq$.

If dimension zero is admitted, $T^{\ast }X=X$ and $\alpha =0$, so every diffeomorphism preserves $\alpha$ and this requested distinction is impossible. The example therefore applies in positive dimension, as required for a nonzero differential one-form.