Riemannian Geometry Part 2

Problem 6.52 Let $(M, \Omega)$ be an almost symplectic manifold. An almost complex structure $J$ on the manifold $M$ is called compatible with the almost symplectic structure $\Omega$, if $g(X, Y)=\Omega(J X, Y)$ is an Hermitian metric on $M$, i.e. (a) $\Omega(J X, X)>0$ for any non-zero tangent vector $X \in T_{p} M, p \in M$. (b) $\Omega(J X, Y)+\Omega(X, J Y)=0$ for any tangent vectors $X, Y \in T_{p} M, p \in M$. Prove that on any almost symplectic manifold $(M, \Omega)$ there exists an almost complex structure $J$ which is compatible with $\Omega$. The relevant theory is developed, for instance, in Gromov [14] and Aebisher et al. [1] .

Problem 6.53 Consider $M=\mathbb{R}^{2} \backslash\{(0,0)\}$ with the usual metric $g=\mathrm{d} x^{2}+\mathrm{d} y^{2}$ and consider the distance function $d_{g}$ given by $$ \begin{aligned} d_{g}: M \times M & \rightarrow \mathbb{R}^{+} \\ (p, q) & \mapsto d_{g}(p, q)=\inf \int_{0}^{1} \sqrt{g\left(\gamma^{\prime}(t), \gamma^{\prime}(t)\right)} \mathrm{d} t, \end{aligned} $$ where $\gamma$ denotes a piecewise $C^{\infty}$ curve with $\gamma(0)=p$ and $\gamma(1)=q$. (i) Compute the distance between $p=(-1,0)$ and $q=(1,0)$. (ii) Is there a geodesic minimizing the distance between $p$ and $q$ ? (iii) Is the topological metric space $\left(M, d_{g}\right)$ complete? (iv) A Riemannian manifold is said to be geodesically complete if every geodesic $\gamma(t)$ is defined for every real value of the parameter $t$. Is in the present case $M$ geodesically complete? (v) Find an open neighbourhood $U_{p}$ for each point $p \in M$, such that for all $q \in U_{p}$, the distance $d_{g}(p, q)$ be achieved by a geodesic.

Problem 6.54 (i) Find an example of a connected Riemannian manifold $(M, g)$ to show that the property "Any $p, q \in M$ can be joined by a geodesic whose arc length equals the distance $d_{g}(p, q)$ " (see Problem 6.53) does not imply that $M$ is complete. (ii) Find an example of a connected Riemannian manifold to show that a minimal geodesic between two points need not be unique; in fact, there may be infinitely many.