nLab geodesic flow

Idea

For $(X,g)$ a Riemannian manifold and $p \in X$ a point, the geodesic flow at $p$ is the map defined on an open neighbourhood of the origin in $(T_p X ) \times \mathbb{R}$ that sends $(v,r)$ to the endpoint of the geodesic that starts with tangent vector $v$ at $p$ and has length $r$.

(…)

Definition

Let $(X,g)$ be a Riemannian manifold

Definition

(…) geodesic flow (…)

The following are some auxiliary definitions that serve to analyse properties of geodesic flow (see Properties).

For $p \in X$ a point and $r \in \mathbb{R}$ a positive real number, we write

$$B_p(r) = \{x \in X | d(p,x) \lt r\} = \{ \exp( v) : T_p X \to X | |v| \lt r \} \subset X$$

for set of points which are of distance less than $r$ away from $p$. As the propositions below assert, for small enough $r$ this is diffeomorphic to an open ball and we speak of metric balls or geodesic balls .

Definition

For $p \in P$ a point, the injectivity radius $inj_p \in \mathbb{R}$ is the supremum over all values of $r \in \mathbb{R}$ such that the geodesic flow starting at $p$ with radius $r$ $\exp(-) : B_r(T_p X) \to X$ is a diffeomorphism onto its image.

The injectivity radius of $(X,g)$ is the infimum of the injectivity radii at each point.

Properties

Properties of the injectivity radius

Proposition

The injectivity radius is

• either equal to half the length of the smalled periodic geodesic,

• or equal to the smallest distance between two conjugate points.

This appears for instance as scholium 91 in (Berger).

Lower bounds on the injectivity radius

There are several lower boundas on the injectivity radius of a Riemannian manifold.

Proposition

The convexity radius is always less than or equal to half of the injectivity radius:

$$conv (X,g) \leq \frac{1}{2} inj(X,g) \,.$$

This appears for instance as proposition IX.6.1 in Chavel, where it is attributed to M. Berger (1976). In (Berger) it is proposition 95.

Let $R$ be the Riemann curvature tensor of $g$. For $p \in X$ the sectional curvature of a plane spanned by vectors $v,w \in T_p X$ is

$$K(v,w) := \frac{R(v,w,v,w)}{g(v,v)g(w,w) - g(v,w)^2} \,.$$

Say that $(X,g)$ is complete if, equivalently,

• with the distance function $X$ is a complete metric space;

• $(X,g)$ is geodesically complete in that for all $v \in T_p X$ the flow $t \mapsto \exp_p(t v)$ exists for all $t \in \mathbb{R}$.

Theorem

Let $(X,g)$ be complete and such that

1. the absolute value of the sectional curvature at all points is bounded from above;

2. the volume of the geodesic unit ball at all points is bounded from below.

Then the injectivity radius is positive.

This is due to (CheegerGromovTaylor). A survey is in (Grant).

Theorem

Every paracompact manifold admits a complete Riemannian metric with

• bounded absolute sectional curvature;

• positive convexity radius

• and hence with positive injectivity radius.

This is shown in (Greene).

References

• Gabriel Paternein, Geodesic flows Birkhäuser (1999)

The following is literature on injectivity radius estimates

A general exposition is in sectin 6 “Injectivity, Convexity radius and cut locuss” of

• Marcel Berger, A panoramic view of Riemannian geometry

Also section IX of

• Isaac Chavel, Riemannian geometry: a modern introduction

A survey of the main estimates is in

• James Grant, Injectivity radius estimates (pdf)

The main theorem is due to

• Jeff Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds , J. Differential Geom., 17 (1982), pp. 15–53.

Older results on compact manifolds are in

• Jeff Cheeger, Finiteness theorems for Riemannian manifolds .

The existence of metrics with all the required propertiers for the injectivity estimates (completeness, bounded absolute sectional curvature) on paracompact manifolds is shown in

• R. Greene, Complete metrics of bounded curvature on noncompact manifolds Archiv der Mathematik Volume 31, Number 1 (1978)

More discussion of construction of Riemannian manifolds with bounds on curvature and volume is in

• John Lott, Zhongmin Chen, Manifolds with quadratic curvature decay and slow volume growth (pdf)

Analogous results for Lorentzian manifolds are discussed in

• Bing-Long Chen, Philippe G. LeFloch, Injectivity Radius of Lorentzian Manifolds (pdf)