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Dear Marco,

If you take unparameterized geodesics (the quotient of the unit co-sphere bundle by the action of the geodesic flow), here is an answer:

The space of geodesics of a Riemannian or even Finsler manifold of dimension $n$ is itself a manifold of dimension $2n - 2$ in a variety of special, but albeit important, cases. For example, the space of geodesics of any rank-one symmetric space (Euclidean space, hyperbolic space, the sphere, real, complex, quaternionic spaces, the Cayley plane, complex hyperbolic space) is a manifold. Other less standard examples are Zoll manifolds, Hadamard manifolds or any convex neighborhood of a Riemannian or Finsler space.

When the space of geodesics is a manifold, it carries a natural symplectic structure inherited by symplectic reduction from the unit co-sphere bundle of the Riemannian or Finsler metric. Families of geodesics normal to immersed submanifolds are immersed Lagrangian submanifolds in the space of geodesics (a result that dates back to Hamilton's Theory of systems of rays, arguably the first paper on symplectic geometry). In particular, points in the manifold correspond to Lagrangian spheres in the space of geodesics (i.e. all geodesics passing through a point is a Lagrangian sphere). Using these spheres and the symplectic structure you can easily reconstruct the metric (see the paper Symplectic geometry and Hilbert's fourth problem, Journal Diff. Geom. Volume 69, Number 2 (2005) for this and as reference to everything else I'm writing here).

To make things concrete here are some examples:

$\bullet$ The space of geodesics of Euclidean or hyperbolic n-space is symplectomorphic to the cotangent of the $n-1$-sphere.

$\bullet$ The space of geodesics of the $n$-sphere or the $n$-dimensional real projective space is symplectomorphic to a complex quadric = the Grassmannian of oriented two-planes in ${\mathbb R}^{n+1}$

$\bullet$ The space of geodesics of the complex projective space ${\mathbb CP}^n$ is symplectomorphic to the space of 1-2 flags (point-line) in ${\mathbb CP}^n$

The space of geodesics may inherit additional structure from the structure of the manifold: for example Hitchin looks at the complex structure of the space of geodesics of ${\mathbb R}^3$ to study monopoles and minimal surfaces in Monopoles and geodesics, Comm. Math. Phys. Volume 83, Number 4 (1982), 579-602.

If you want additional references go to my JDG paper or my 1995 thesis (The symplectic geometry of spaces of geodesics) that you will find on the web.

1

Dear Marco,

If you take unparameterized geodesics (the quotient of the unit co-sphere bundle by the action of the geodesic flow), here is an answer:

The space of geodesics of a Riemannian or even Finsler manifold of dimension $n$ is itself a manifold of dimension $2n - 2$ in a variety of special, but important cases. For example, the space of geodesics any rank-one symmetric space (Euclidean space, hyperbolic space, the sphere, real, complex, quaternionic spaces, the Cayley plane, complex hyperbolic space) is a manifold. Other less standard examples are Zoll manifolds, Hadamard manifolds or any convex neighborhood of a Riemannian or Finsler space.

When the space of geodesics is a manifold, it carries a natural symplectic structure inherited by symplectic reduction from the unit co-sphere bundle of the Riemannian or Finsler metric. Families of geodesics normal to immersed submanifolds are immersed Lagrangian submanifolds in the space of geodesics (a result that dates back to Hamilton's Theory of systems of rays, arguably the first paper on symplectic geometry). In particular, points in the manifold correspond to Lagrangian spheres in the space of geodesics (i.e. all geodesics passing through a point is a Lagrangian sphere). Using these spheres and the symplectic structure you can easily reconstruct the metric (see the paper Symplectic geometry and Hilbert's fourth problem, Journal Diff. Geom. Volume 69, Number 2 (2005) for this and as reference to everything else I'm writing here).

To make things concrete here are some examples:

$\bullet$ The space of geodesics of Euclidean or hyperbolic n-space is symplectomorphic to the cotangent of the $n-1$-sphere.

$\bullet$ The space of geodesics of the $n$-sphere or the $n$-dimensional real projective space is symplectomorphic to a complex quadric = the Grassmannian of oriented two-planes in ${\mathbb R}^{n+1}$

$\bullet$ The space of geodesics of the complex projective space ${\mathbb CP}^n$ is symplectomorphic to the space of 1-2 flags (point-line) in ${\mathbb CP}^n$

The space of geodesics may inherit additional structure from the structure of the manifold: for example Hitchin looks at the complex structure of the space of geodesics of ${\mathbb R}^3$ to study monopoles and minimal surfaces in Monopoles and geodesics, Comm. Math. Phys. Volume 83, Number 4 (1982), 579-602.

If you want additional references go to my JDG paper or my 1995 thesis (The symplectic geometry of spaces of geodesics) that you will find on the web.