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\begin{document}

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\section*{geodesic flow}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{Properties}{Properties}\dotfill \pageref*{Properties} \linebreak
\noindent\hyperlink{properties_of_the_injectivity_radius}{Properties of the injectivity radius}\dotfill \pageref*{properties_of_the_injectivity_radius} \linebreak
\noindent\hyperlink{lower_bounds_on_the_injectivity_radius}{Lower bounds on the injectivity radius}\dotfill \pageref*{lower_bounds_on_the_injectivity_radius} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

For $(X,g)$ a [[Riemannian manifold]] and $p \in X$ a point, the \textbf{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 $r$.

(\ldots{})

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\begin{udefn}
(\ldots{}) geodesic flow (\ldots{})

\end{udefn}
The following are some auxiliary definitions that serve to analyse properties of geodesic flow (see \hyperlink{Properties}{Properties}).

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

\begin{displaymath}
B_p(r) = \{x \in X | d(p,x) \lt r\}  = \{ \exp( v) : T_p X \to X | |v| \lt r \} \subset X
\end{displaymath}
for set of points which are of [[distance]] less than $r$ away from $p$. As the propositions \hyperlink{Properties}{below} assert, for small enough $r$ this is [[diffeomorphic]] to an [[open ball]] and we speak of \emph{metric balls} or \emph{geodesic balls} .

\hypertarget{definition_3}{}\paragraph*{{Definition}}\label{definition_3}

For $p \in P$ a point, the \textbf{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.

\hypertarget{Properties}{}\subsection*{{Properties}}\label{Properties}

\hypertarget{properties_of_the_injectivity_radius}{}\subsubsection*{{Properties of the injectivity radius}}\label{properties_of_the_injectivity_radius}

\begin{uproposition}
The injectivity radius is

\begin{itemize}%
\item either equal to half the length of the smalled periodic geodesic,


\item or equal to the smallest distance between two conjugate points.



\end{itemize}
\end{uproposition}
This appears for instance as scholium 91 in (\hyperlink{Berger}{Berger}).

\hypertarget{lower_bounds_on_the_injectivity_radius}{}\subsubsection*{{Lower bounds on the injectivity radius}}\label{lower_bounds_on_the_injectivity_radius}

There are several lower boundas on the \hyperlink{InjectivityRadius}{injectivity radius} of a Riemannian manifold.

\begin{uproposition}
The [[convexity radius]] is always less than or equal to half of the injectivity radius:

\begin{displaymath}
conv (X,g) \leq \frac{1}{2} inj(X,g)
  \,.
\end{displaymath}
\end{uproposition}
This appears for instance as proposition IX.6.1 in \hyperlink{Chavel}{Chavel}, where it is attributed to M. Berger (1976). In (\hyperlink{Berger}{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

\begin{displaymath}
K(v,w) :=
  \frac{R(v,w,v,w)}{g(v,v)g(w,w) - g(v,w)^2}
  \,.
\end{displaymath}
Say that $(X,g)$ is \textbf{complete} if, equivalently,

\begin{itemize}%
\item with the [[distance]] function $X$ is a [[complete metric space]];


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



\end{itemize}
\begin{utheorem}
Let $(X,g)$ be complete and such that

\begin{enumerate}%
\item the [[absolute value]] of the sectional curvature at all points is bounded from above;


\item the [[volume]] of the geodesic unit ball at all points is bounded from below.



\end{enumerate}
Then the injectivity radius is positive.

\end{utheorem}
This is due to (\hyperlink{CheegerGromovTaylor}{CheegerGromovTaylor}). A survey is in (\hyperlink{Grant}{Grant}).

\begin{utheorem}
Every [[paracompact manifold]] admits a complete [[Riemannian metric]] with

\begin{itemize}%
\item bounded absolute sectional curvature;


\item positive convexity radius


\item and hence with positive injectivity radius.



\end{itemize}
\end{utheorem}
This is shown in (\hyperlink{Greene}{Greene}).

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item Gabriel Paternein, \emph{Geodesic flows} Birkh\"a{}user (1999)

\end{itemize}
The following is literature on injectivity radius estimates

A general exposition is in sectin 6 ``Injectivity, Convexity radius and cut locuss'' of

\begin{itemize}%
\item Marcel Berger, \emph{A panoramic view of Riemannian geometry}

\end{itemize}
Also section IX of

\begin{itemize}%
\item Isaac Chavel, \emph{Riemannian geometry: a modern introduction}

\end{itemize}
A survey of the main estimates is in

\begin{itemize}%
\item James Grant, \emph{Injectivity radius estimates} (\href{http://www.mat.univie.ac.at/~grant/papers/talk.pdf}{pdf})

\end{itemize}
The main theorem is due to

\begin{itemize}%
\item [[Jeff Cheeger]], M. Gromov, and M. Taylor, \emph{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.

\end{itemize}
Older results on compact manifolds are in

\begin{itemize}%
\item [[Jeff Cheeger]], \emph{Finiteness theorems for Riemannian manifolds} .

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

\begin{itemize}%
\item R. Greene, \emph{Complete metrics of bounded curvature on noncompact manifolds} Archiv der Mathematik Volume 31, Number 1 (1978)

\end{itemize}
More discussion of construction of Riemannian manifolds with bounds on curvature and volume is in

\begin{itemize}%
\item John Lott, Zhongmin Chen, \emph{Manifolds with quadratic curvature decay and slow volume growth} (\href{http://math.berkeley.edu/~lott/science.pdf}{pdf})

\end{itemize}
Analogous results for [[Lorentzian manifold]]s are discussed in

\begin{itemize}%
\item Bing-Long Chen, Philippe G. LeFloch, \emph{Injectivity Radius of Lorentzian Manifolds} (\href{http://philippelefloch.files.wordpress.com/2009/12/2008-chenlefloch-cmp.pdf}{pdf})

\end{itemize}
[[!redirects injectivity radius]]



\end{document}