Questions tagged [geodesics]

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Smooth manifolds for which every metric is geodesically convex

Are there non compact smooth manifolds which have the property that every Riemannian metric is geodesically convex? Note that a manifold for which every Riemannian metric is complete must be compact. ...
Asaf Shachar's user avatar
  • 6,611
34 votes
0 answers
692 views

Metrics on the 3-sphere with knotted geodesics

According to answers to this question every metrics on $S^3$ admits a simple closed geodesic. Given a knot (or link) $K$, it's also quite simple to build a metric on $S^3$ such that $K$ is a geodesic (...
Marco Golla's user avatar
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3 votes
0 answers
174 views

Does null geodesic flow live on a natural compact bundle?

Let $(M,g)$ be a compact pseudo-Riemannian manifold (closed or with boundary). A geodesic $\gamma:(a,b)\to M$ is called null if $g_{ij}\dot\gamma^i\dot\gamma^j=0$. The geodesic flow can be seen as a ...
Joonas Ilmavirta's user avatar
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 (...
Joseph O'Rourke's user avatar
4 votes
1 answer
363 views

Null tetrad transformation

I have been going through the Chandrasekhar's "The Mathematical Theory of Black holes", in particular the chapter on Newman Penrose formalism. I have a question about what he calls a "class III ...
GregVoit's user avatar
  • 475
2 votes
0 answers
147 views

Derivation of gradient of SSE in Geodesic Regression

On page 79 (or page 5) of this this paper the gradient of the SSE of the Geodesic model is described explicitly. My question is how are these equitations derived in detail; where can I find the ...
ABIM's user avatar
  • 5,019
2 votes
1 answer
333 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, ...
Joseph O'Rourke's user avatar
1 vote
1 answer
270 views

Obstruction to the existence of global isometries on a constant-curvature Riemannian manifold

Let $M$ be an $m$-dimensional simply connected Riemannian manifold that is not geodesically complete. Suppose $M$ has constant sectional curvature. Because the curvature is constant, locally $M$ ...
Josh Burby's user avatar
4 votes
1 answer
766 views

The heat kernel as an exponential of an integral

In $\mathbb{R}^n$, if $\gamma$ is a line segment between $x_0 = \gamma (0)$ and $x = \gamma (t)$, one has the following formula: $$\frac {\mathbb{e}^{- \frac{1}{4} \int_0^t <\dot{\gamma}, \dot{\...
Alex M.'s user avatar
  • 5,207
7 votes
1 answer
927 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 ...
Joseph O'Rourke's user avatar
2 votes
1 answer
128 views

Geodesic paths on a flat sphere

Let $S$ be a $2$-dimensional sphere endowed with a flat metric with $3$ conical singularities of positive curvature. Typically, $S$ is a metric space you get when you glue two copies of the same ...
Selim G's user avatar
  • 2,626
4 votes
1 answer
163 views

Geodesic comparison in Hadamard space

Let $X$ be a hadamard space and $\gamma_1, \gamma_2 \colon \mathbb{R}\rightarrow X$ be two geodesics. Part 2 of Coroallary 2.5 in http://www.iam.uni-bonn.de/fileadmin/WT/Inhalt/people/Karl-...
user35593's user avatar
  • 2,286
2 votes
0 answers
279 views

Log of heat kernel for positive time

A well-known theorem by Varadhan relates the logarithm of the heat kernel on a manifold and the geodesic distance function. In particular, if $d(x,y)$ is geodesic distance from $x$ to $y$ and $p(t,x,...
Justin's user avatar
  • 695
6 votes
1 answer
348 views

Tori in Compact Riemannian Symmetric Spaces

The closure of a 1-parameter subgroup in a compact Lie group is a torus. To what extent does this result generalize to compact Riemannian symmetric spaces? In other words, is the closure of a geodesic ...
Oliver Jones's user avatar
  • 1,368
18 votes
2 answers
1k views

Simple, closed geodesics in $\mathbb{S}^3$ manifold

Lyusternik and Shnirel'man were the first to prove Poincaré's conjecture that any Riemannian metric on $\mathbb{S}^2$ has at least three simple (non-self-intersecting), closed geodesics. See, e.g., p....
Joseph O'Rourke's user avatar
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 ...
Joseph O'Rourke's user avatar
2 votes
1 answer
302 views

2 questions about loops and negative curvature

$(M^n,g)$ is a compact $n$ dimensional manifold of negative curvature with n>2 . let $\alpha$ be a simple closed geodesic loop in $M$ based at a point $p$ 1) will the geodesic in the free homotopy ...
user avatar
10 votes
4 answers
1k views

Geodesic path on the unit sphere with the sup norm

Let $X$ be the unit sphere of $\mathbb{R}^n$ with the sup norm, i.e. $X=\{x\in\mathbb{R}^n: \|x\|_{\infty}=1\}$. Let the metric $d$ on $X$ be the geodesic metric induced by the sup norm, i.e for any ...
TCL's user avatar
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