straight lines on closed manifolds
bo198214
another forum "If one always flies in one direction (in our universe)
does one then return to the starting point?" I was asking myself
whether there are closed manifolds with points, if one starts flying in
*any* initial direction from that point in a straight line, one never
returns to the starting point.
I could construct an example on the torus, such that exactly for
rational slopes of the straight line one returns to the starting point.
Whats at all the name of such a straight line that returns to the
starting point, and such a straight line that doesnt, can one say
closed and non-closed straight line?
[ Moderator's note: I believe the term the poster is looking for is
"geodesic loop". Thus the question is whether there is a closed
Riemannian manifold with a point that is not the start of a geodesic
loop.
-RI ]
Ryan Budney
bo198214 wrote:
> Am not really familiar with manifolds, but based on the question in
> another forum "If one always flies in one direction (in our universe)
> does one then return to the starting point?" I was asking myself
> whether there are closed manifolds with points, if one starts flying in
> *any* initial direction from that point in a straight line, one never
> returns to the starting point.
Let f : [a,b] --> M be a geodesic meaning the covariant derivative of
f' is zero, ie a "straight" line. We call f(a) the start point of f
and f(b) the end point. If you formulate your question in terms of
looking for geodesics f such that f(a)=f(b) with a not equal to b, then
every point x of M has such a geodesic f with f(a)=x. The idea is that
every closed manifold M has an injectivity radius, meaning a minimal
positive number R so that any geodesic of length less than R is
one-to-one. So there must exist a geodesic path of length exactly the
injectivity radius with f(a)=f(b) but on any proper subinterval of
[a,b], f is injective. (in most places the injectivity radius is
defined to be half R, I think)
If you demand also that f'(a)=f'(b) then you have a different story.
For example, in hyperbolic manifolds, its known that every element of
the fundamental group of the manifold has a unique closed geodesic
representative (with f'(a)=f'(b)). But the fundamental group is
countably generated, so not every point of M can have such a geodesic
passing through it, provided the dimension of M is 2 or larger. So I
guess the simplest example of a manifold with the properties you're
looking for would be a closed hyperbolic surface of genus 2.
bo198214
Already the first part quite satisfies me :) Will read a bit more about
this injectivity radius.
Ryan Budney
Re-reading my post I realize my answer wasn't as thorough as I would
like. The first part doesn't actually answer your question because I
mis-state the definition of injectivity radius. The injectivity radius
is the smallest positive number R so that if you follow a geodesic of
length less than R in any direction v from some point x, then you do
not cross the path of any other geodesic of length less than R starting
at x in some direction v' (for any x, v, v'). So the result I mention
about injectivity radius at best gives you a path that starts at x,
then at some point changes direction and eventually returns to x (so if
you think of this path as a closed loop, it has at most two changes of
direction in it -- you're interested in ones with at most one change of
direction, the starting/end point).
Anyhow, that made me realize your question is more difficult than I
first thought. But it has been answered in the literature. In the
math literature, what you're asking would be stated as "is there a
compact riemann manifold with no closed geodesic loops?" The answer is
no and was apparently answered a long time ago by Lusternik and Fet
(although I haven't been able to get my hands on their papers). There
is a nice modern alternative though. Regina Rotman proved a pretty
neat looking theorem: if q is the smallest integer so that the q-th
homotopy group of your manifold M is non-trivial, then for any point x
in the manifold there is a geodesic loop starting and ending at x with
length less than or equal to twice q times the diameter of the
manifold. The "diameter" means the furthest distance between two
points in the manifold, where distance is measured by the shortest
geodesic between the two points. And "geodesic loop" is a geodesic
that starts and ends at the same point, but the tangent vectors may not
agree. For a closed manifold M, q is always some number less than or
equal to the dimension of M (and greater than or equal to 1) -- in
general, it's not an easy number to compute.
Anyhow, I hope that helps and sorry about the earlier goof-up.
bo198214
Ryan Budney wrote:
> math literature, what you're asking would be stated as "is there a
> compact riemann manifold with no closed geodesic loops?" The answer is
> no and was apparently answered a long time ago by Lusternik and Fet
loop?
Same with injectivity radius, it would only mean that there is a point,
with two intersecting geodesics, wouldnt it?
Fortunately the result of Regina Rotman anwers shows for every point
the existence of a geodesic loop.