All Questions
63 questions
4
votes
1
answer
137
views
Geodesic-like curves stemming from the heat kernel on a manifold
Consider a smooth $n$-dimensional Riemannian manifold $M$ with sufficiently nice geometric and topological properties such that there exist a unique heat kernel $(t,x,y) \mapsto p(t,x,y)$ on it ($t>...
CommunityBot
- 1
2
votes
1
answer
80
views
Relation between the geodesics of Finsler norms $F(V)$ and $F(-V)$
I am trying to solve this exercise. Let $(M,F)$ be a Finsler space and define $\tilde{F}(x,y):=F(x,-y)$. Then $(M,\tilde{F})$ is a Finsler space and given a geodesic $t\mapsto \gamma(t)$ of $F$, $t\...
- 227
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-...
- 4,312
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
1
vote
1
answer
98
views
Polar coordinates of a set with different radius and angle
Let $M$ be a $2$-dimensional Riemannian manifold and let $U\subset M$ be an open set. Suppose there exist polar coordinates $(r,\theta)$ with center $q\in M$ such that
$$U=\lbrace{ (r,\theta): 0<...
- 45k
15
votes
2
answers
1k
views
When is a flow geodesic and how to construct the connection from it
Let $(M,\Gamma)$ be a $C^\infty$ $n$ dimensional real manifold with a linear connection $\Gamma$ on it. I know the following:
If $\gamma:[t_0,t_1]\rightarrow M$ is a smooth curve and is a geodesic, ...
- 108k
5
votes
1
answer
321
views
Simply connected manifolds with dense geodesics on the tangent bundle
A unit speed geodesic $\gamma:I\to M$ on a Riemannian manifold $M$ can be lifted to a curve $\sigma=(\gamma,\gamma')$ on $SM$, the unit sphere bundle (the unit tangent bundle) of $M$.
Let us say that ...
CommunityBot
- 1
6
votes
1
answer
300
views
Geodesics for non differentiable riemannian metric
Let $M$ be a differentiable manifold of dimension $n>2$ with a Riemannian metric $g=\sum_{i,j=1}^ng_{ij}dx_idx_j$ such that in some points on $M$ its coefficients $g_{ij}$ are not differentiable (...
- 45k
4
votes
1
answer
161
views
Does every Zoll metric on $\mathbb{S}^2$ arise from a perturbation of the round metric?
The introduction here states 'A formal perturbation argument
of Funk later indicated that, modulo isometries and rescalings, the general Zoll
metric on $\mathbb{S}^2$ depends on one odd function $f:\...
- 108k
2
votes
0
answers
144
views
Flow on invariant Lagrangian tori
The most concrete version of the question is :
A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...
- 13.5k
3
votes
0
answers
178
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 ...
- 8,086
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, ...
- 108k
4
votes
1
answer
797
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{\...
- 5,407