All Questions
13 questions
23
votes
2
answers
1k
views
Can we make distances in a finite subset of a manifold whatever we want?
Given a connected smooth manifold $M$ of dimension $m>1$, points $p_1,\dots,p_n\in M$ and positive values $\{d_{i,j};1\leq i<j\leq n\}$ satisfying the strict triangle inequalities $d_{i,j}<d_{...
- 10.6k
3
votes
0
answers
531
views
Geodesics (Local vs Global)
Let $M$ be a Riemannian manifold, and $p,q\in M$ two points. Now, if $M$ is a complete metric space the Hopf–Rinow theorem ensures that there is a geodesic $C\subset M$ joining $p$ and $q$ that ...
user75933
7
votes
2
answers
434
views
Convexity in co-ordinate charts of geodesic balls
Let $g_{ij}$ be a Riemannian metric tensor on an open subset $U\subseteq \mathbb{R}^n$, and let $p\in U$.
I would guess the following is true:
for $\epsilon$ sufficiently small, the $g$-geodesic ...
- 3,212
13
votes
1
answer
550
views
Regularity of geodesics
If $M$ is a graph of a $C^1$ function $f:\mathbb{R}^n\to\mathbb{R}$, is it true that the length minimizing geodesics on $M$ are $C^1$? I expect a counterexample.
For a related discussion see Metric ...
- 28k
12
votes
3
answers
988
views
Primary definition of a geodesic
I am wondering if there is a sense in which one of these definitions
for a geodesic on a smooth Riemannian manifold is primary to the other.
A geodesic has acceleration zero, i.e., it is self-...
- 151k
5
votes
0
answers
464
views
Examples of spiraling geodesics?
Does there exist a closed, bounded surface $S$ embedded in $\mathbb{R}^3$
that has a geodesic $\gamma$ that spirals around a point $x$, getting closer
and closer, but never reaching $x$?
Here I ...
- 151k
11
votes
3
answers
667
views
Which surfaces admit unbounded-length simple geodesics?
Let $S$ be a surface embedded in $\mathbb{R}^3$.
A simple geodesic on $S$ is one that does not self-intersect.
Some surfaces have simple geodesics whose length exceeds any
given bound $L$. For example,...
- 151k
2
votes
2
answers
496
views
Geodesic on Banach Manifold [closed]
Is there a way of defining a geodesic on a Banach Manifold $M$ which is not itself a Hilbert Manifold?
- 5,405
6
votes
1
answer
184
views
Self-avoiding/reflecting geodesics on a convex surface
Let $S$ be the surface of a convex body embedded in $\mathbb{R}^3$.
For me $S$ is a convex polyhedron,
but I am happy to view $S$ as a smooth body with positive Gaussian curvature
at each point, or ...
- 151k
29
votes
2
answers
1k
views
Is every closed curve in 3D a geodesic on a genus-0 surface?
Let $\gamma$ be a smooth, closed, unknotted curve embedded in $\mathbb{R}^3$.
Q. Does there always exist a smooth, embedded, genus-zero surface
$S \subset \mathbb{R}^3$
such that $\gamma$ is a (...
- 151k
2
votes
1
answer
346
views
Closed geodesics that cross one another frequently
Let $S$ be a smooth, closed, genus zero surface in $\mathbb{R}^3$.
$S$ has at least three simple (non-self-intersecting), closed geodesics by
a theorem of Lyusternik and Shnirel'man.
Alternatively, ...
- 151k
7
votes
1
answer
962
views
Which surfaces have only a finite number of connecting geodesics?
Q1. For a smooth, closed (compact) surface $S$ embedded in $\mathbb{R}^3$,
under which conditions is it true that, for every pair of points
$a,b \in S$, there are an infinite number of ...
- 151k
39
votes
5
answers
3k
views
Surfaces filled densely by a geodesic
Which smooth, closed surfaces $S \subset \mathbb{R}^3$ have no
single geodesic $\gamma$ that fills $S$ densely?
Say a geodesic $\gamma$ "fills $S$ densely" if the closure of the set of points
...
- 151k