\( \def\CC{{\mathbb C}} \def\RR{{\mathbb R}} \def\NN{{\mathbb N}} \def\ZZ{{\mathbb Z}} \def\TT{{\mathbb T}} \def\CF{{\operatorname{CF}^\bullet}} \def\HF{{\operatorname{HF}^\bullet}} \def\SH{{\operatorname{SH}^\bullet}} \def\ot{{\leftarrow}} \def\st{\;:\;} \def\Fuk{{\operatorname{Fuk}}} \def\emprod{m} \def\cone{\operatorname{Cone}} \def\Flux{\operatorname{Flux}} \def\li{i} \def\ev{\operatorname{ev}} \def\id{\operatorname{id}} \def\grad{\operatorname{grad}} \def\ind{\operatorname{ind}} \def\weight{\operatorname{wt}} \def\Sym{\operatorname{Sym}} \def\HeF{\widehat{CHF}^\bullet} \def\HHeF{\widehat{HHF}^\bullet} \def\Spinc{\operatorname{Spin}^c} \def\min{\operatorname{min}} \def\div{\operatorname{div}} \def\SH{{\operatorname{SH}^\bullet}} \def\CF{{\operatorname{CF}^\bullet}} \def\Tw{{\operatorname{Tw}}} \def\Log{{\operatorname{Log}}} \def\TropB{{\operatorname{TropB}}} \def\wt{{\operatorname{wt}}} \def\Span{{\operatorname{span}}} \def\Crit{\operatorname{Crit}} \def\into{\hookrightarrow} \def\tensor{\otimes} \def\CP{\mathbb{CP}} \def\eps{\varepsilon} \) SympSnip:

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  1. Let \(G:TQ\to T^*Q\) be the isomorphism induced by \(g\). Define \(g^*:T^*Q\times T^*Q \to \RR\) as \(g^*:=g\circ (G^{-1}\times G^{-1})\). Note that in local coordinates \(g^*_{ij}=g^{ij}\), where \((g^{ij})\) is the inverse of the local expression \((g_{ij})\) of \(g\). Consider now the Hamiltonian \(H(q,p):=g_q^*(p,p)\) on \(T^*Q\). Write \(g^*(p,p')=\sum_{i,j=1}^n g^*_{ij}p_i p'_j=\sum_{i,j=1}^n g^{ij}p_i p_j\) in local coordinates. We compute \begin{align*} V_g&=J\nabla H\\ &=J\nabla\left(\sum_{i,j=1}^n g^{ij}p_ip_j\right)\\ &=J\sum_{i,j=1}^n\left(\sum_{k=1}^n\frac{\partial g^{ij}}{\partial q_k} p_ip_j\partial_{q_k}+2g^{ij}p_i\partial_{p_j}\right)\\ &=\sum_{i,j=1}^n\left(\sum_{k=1}^n\frac{\partial g^{ij}}{\partial q_k} p_ip_j\partial_{p_k}-2g^{ij}p_i\partial_{q_j}\right) \end{align*}
  2. Note the projected curve \(\gamma=\pi_Q(\tilde\gamma)\) satisfies the differential equation \(\dot\gamma_i=-2\sum_{j=1}^n g^{ij}p_j\) (equivalently, \(p_i=-\frac{1}{2}\sum_{j=1}^n g_{ij} \dot \gamma_j\)) and furthermore \(\dot p_j=\sum_{k,l=1}^n \frac{\partial g^{lk}}{\partial q_j}p_kp_l=\frac{1}{4}\sum_{k,l=1}^n\frac{\partial g_{kl}}{\partial q_j}\dot\gamma_k\dot\gamma_l\). Therefore \begin{align*} \ddot\gamma_i&=-2\sum_{j=1}^n \left(\sum_{k=1}^n \frac{\partial g^{ij}}{\partial q_k}p_j\dot\gamma_k+g^{ij}\dot p_j\right)\\ &=\sum_{j,k,l=1}^ng^{ij}\left(\frac{\partial g_{jl}}{\partial q_k}\dot\gamma_l\dot\gamma_k-\frac{1}{2}\frac{\partial g_{kl}}{\partial q_j}\dot\gamma_k\dot\gamma_l\right)\\ \end{align*} which is precisely the geodesic equation obtained from minimizing the action.
  3. The second page of the spectral sequence is given by \(E^2_{p,q}=H_p(S^n;H_q(\Omega S^n;\ZZ))\cong H_p(S^n;\ZZ)\otimes H_q(\Omega S^n;\ZZ)\), which is non-zero only for \(p=0,n\). Note that since \(PS^n\) is contractible the spectral sequence must converge to zero (except at the origin). Therefore:
    • No differential can hit the terms \(E^n_{0,i}\cong H_i(\Omega S^n;\ZZ)\) for \(i=1,\dots,n-2\), thus they must vanish.
    • The edge homomorphisms \(d^n:E^n_{n,q}\to E^n_{n,q+n-1}, q\geq0\) must be isomorphisms. This gives \(H_i(\Omega S^n;\ZZ)\cong E^n_{n,i}\cong E^n_{0,i+n-1}\cong H_{i+n-1}(\Omega S^n;\ZZ)\), so that the first \(n-1\) homology groups determine the rest.
    We conclude \(H_*(\Omega S^n;\ZZ)\cong \ZZ[x]\) with \(|x|=n-1\) as required.
  4. By the previous part, we now have \(E^2_{p,q}=\ZZ\) for \(p=0,n\) and \(q=0,n-1,2(n-1),\dots\) and \(0\) otherwise. In particular, the edge homomorphisms \(d^n:E^n_{n,i}\to E^n_{0,i+n-1}, i=0,n-1,2(n-1),\dots\) are the only non-trivial differentials in the spectral sequence, so \(E^{n+1}_{p,q}=E^\infty_{p,q}\). These maps are endomorphisms of \(\ZZ\) and thus given by multiplication by \(k\), leaving only terms of the form \((E^{n+1}_{n,i},E^{n+1}_{0,i+n-1})=(\ZZ,\ZZ)\) if \(k=0\) or \((E^{n+1}_{n,i},E^{n+1}_{0,i+n-1})=(0,\ZZ/k\ZZ)\) if \(k\neq 0\) for the next --- and last --- page of the spectral sequence. Therefore, \(\ZZ\) factors appear in pairs, and since \(H_0(LS^n)=\ZZ\) there must be an odd-number of them, excluding the possibility that \(H_*(LS^n)=\ZZ^2\). Lastly, note that since \(H_0(LS^n)=\ZZ\), the first and second possibilities in the question are already excluded.
  5. Let \(H(q,p):=g^*_q(p,p)\) be the Hamiltonian used in part 1. Then \(H\) can't have any non-constant periodic orbits, as otherwise their projection to \(S^n\) would be closed geodesics. If there existed a metric with no closed geodesics, then by the previous part the Hamiltonian Floer cochain complex \(CF(H)\) would have only two generators. Denoting their degrees by \(p\) and \(q\), the possible scenarios are:
    • If \(|p-q|=0\) or \(|p-q|\geq 2\), then there is no differential and \(SH^\bullet(T^*S^n)\cong\ZZ^2\)
    • If \(|p-q|=1\), then the differential is given by multiplication by \(k\) for some \(k\in \ZZ\), and we get \(SH^\bullet(T^*S^n)=\ZZ^2\) if \(k=0\) or \(SH^\bullet(T^*S^n)=\ZZ/k\ZZ\) if \(k\neq0\).
    Viterbo's theorem states that \(SH^\bullet(T^*S^n)\cong H_{-\bullet}(LS^n)\), so all these cases contradict what was proven in part 4. Therefore such metric cannot exist.