exercise 0.0.1

1. Describe a metric on $g^{\ast }:T_q^{\ast }Q\to T_q^{\ast }Q$ on the cotangent bundle. In local coordinates $(q,p)$ for $T^{\ast }Q$, write down the Hamiltonian vector field $V_g$ associated to the Hamiltonian $H(q,p):=g_q^{\ast }(p,p):T_q^{\ast }Q\to \mathbb{R}$.
2. Recall that a geodesic on $Q$ is a curve $\gamma :\mathbb{R}\to Q$ which is locally length minimizing. In particular, it is a minimizer for the action $$E(\gamma )=\frac{1}{2}{\int }_a^b{g}_{\gamma (t)}(\dot{\gamma }(t),\dot{\gamma }(t))dt$$ for every $a,b\in \mathbb{R}$. Prove that if $\hat{\gamma }:\mathbb{R}\to T^{\ast }Q$ is a flow-line of $V_g$, that $\gamma :={\pi }_Q(\hat{\gamma })$ is a geodesic.
3. Let $n>1$. Use the Serre spectral sequence for the fibration $$\mathrm{\Omega }{S}^n\to P{S}^n\to S^n$$ to conclude that $H_{\ast }(\mathrm{\Omega }{S}^n)\simeq \mathbb{Z}[x]$. Here, $\mathrm{\Omega }M$ is the based loop space of $M$, and $PM$ is the based path space of $M$. The element $x$ is of degree $n-1$ and is determined by $$H_n(S^n)\to {\pi }_n(S^n)\to {\pi }_{n-1}(\mathrm{\Omega }{S}^n)\to H_{n-1}(\mathrm{\Omega }{S}^n).$$
4. Let $n>1$. Using the Serre spectral sequence for the fibration $$\mathrm{\Omega }{S}^n\to L{S}^n\to S^n$$ show that as a vector space $H_{\ast }(L{S}^n)\ne 0,\mathbb{Z}/n\mathbb{Z}$ or ${\mathbb{Z}}^2$. Here, $LM$ is the free loop space of $L$.
5. Show that if there exists a metric $g$ for $S^n$ which has no closed geodesics, that there exists a Hamiltonian $H:S^n\to \mathbb{R}$ which has no non-constant orbits, and two constant orbits. Conclude that every metric on $S^n$ has at least one closed geodesic.

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