# nLab geodesic flow

Contents

### Context

#### Riemannian geometry

Riemannian geometry

## Applications

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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

• 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;

• and hence with positive injectivity radius.

This is shown in (Greene).

• 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)