Quasi-geodesic

In mathematics , quasigeodesics (also quasi-geodesics ) occur in differential geometry , metric geometry and geometric group theory. These are curves that are not necessarily the shortest links, but whose length only deviates from the shortest links in a controlled manner.

definition

Let it be a metric space and a (finite or infinite) closed interval. A (not necessarily continuous ) mapping ${\displaystyle (X,d)}$ ${\displaystyle I\subset \mathbb {R} }$

${\displaystyle c\colon I\to X}$

is a quasigeodesic if there are constants such that for all of them : ${\displaystyle \lambda ,\epsilon >0}$ ${\displaystyle s,t\in I}$

${\displaystyle {\frac {1}{\lambda }}|s-t|-\epsilon \leq d(c(s),c(t))\leq \lambda |s-t|+\epsilon }$.

In other words: is a quasi-isometric embedding . ${\displaystyle c\colon I\to X}$

Examples

• In or more generally in every simply connected Riemann manifold of non-positive sectional curvature , a geodesic is always a quasigeodesic. ${\displaystyle \mathbb {R} ^{n}}$
• The logarithmic spiral is a quasigeodesic because it holds . ${\displaystyle c(t)=(t\cos(\ln(1+t)),t\sin(\ln(1+t)))}$ ${\displaystyle |s-t|\leq d(c(s),c(t))\leq (2\pi +1)|s-t|}$
• Controlled disturbances of a quasigeodesics are again quasigeodesics, with a possibly different constant . ${\displaystyle \epsilon }$

More precisely: If there is a quasigeodesic and holds for one constant and all , then it is a quasigeodesic. ${\displaystyle c}$ ${\displaystyle d(c(t),c^{\prime }(t))<K}$ ${\displaystyle K}$ ${\displaystyle t\in I}$ ${\displaystyle c^{\prime }}$

• If there is a quasi-geodesic and a quasi-isometric embedding, then is a quasigeodesic. ${\displaystyle c\colon I\to X}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle f\circ c}$

Morse lemma

Let it be a Gromov hyperbolic space, for example a simply connected Riemannian manifold with negative section curvature . Then every quasigeodesic has a finite Hausdorff distance from the (unique) geodesic through and . ${\displaystyle (X,d)}$ ${\displaystyle c\colon \left[a,b\right]\to X}$ ${\displaystyle c^{\prime }}$ ${\displaystyle c(a)}$ ${\displaystyle c(b)}$

More precisely: There is one for all of them , so that every quasigeodesic in a hyperbolic space lies at a distance from a geodesic. ${\displaystyle \lambda ,\epsilon ,\delta }$ ${\displaystyle R(\lambda ,\epsilon ,\delta )}$ ${\displaystyle (\lambda ,\epsilon )}$ ${\displaystyle \delta }$ ${\displaystyle <R}$

In particular, if -hyperbolic and continuous and rectifiable , then applies to all ${\displaystyle X}$ ${\displaystyle \delta }$ ${\displaystyle c}$ ${\displaystyle x\in im(c^{\prime })}$

${\displaystyle d(x,im(c))\leq \delta |\log _{2}l(c)|+1}$,

where denotes the length of . ${\displaystyle l(c)}$ ${\displaystyle c}$

The analogous statement for CAT (0) -spaces or manifolds of non-positive intersection curvature does not apply. A counterexample is the logarithmic spiral im . ${\displaystyle \mathbb {R} ^{2}}$

literature

• Ghys, Étienne; de la Harpe, Pierre: Quasi-isométries et quasi-géodésiques. Sur les groupes hyperboliques d'après Mikhael Gromov (Bern, 1988), 79-102, Progr. Math., 83, Birkhauser Boston, Boston, MA, 1990.

Web links

• Lück: Hyperbolic Groups
• Wilton: Quasi-geodesics