All Questions
63 questions
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53
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The Frobenius integrability of distrbution and Hyers–Ulam–Rassias stability
Let $M$ be a compact Riemannian manifold. The norm of vector fields are computed with respect to the metric. Moreover for every distribution $D$, the orthogonal projection on $D$ is ...
- 356
3
votes
2
answers
236
views
Lengths of closed geodesics and geodesic segments
Let $M$ be a closed Riemannian manifold of dimension $n \geq 2$. I am looking at sufficient conditions for the following two properties:
existence of closed geodesics of arbitrarily long length on $M$...
- 31
4
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0
answers
246
views
Dynamical obstruction for a vector field to have a Harmonic divergence
Let $(M,g)$ be an analytic Riemannian manifold and $X$ be an analytic vector field on $M$. Can we always have a volume form $\Omega$ such that $\operatorname{Div}_{\Omega} X$ is a harmonic ...
- 356
0
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0
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85
views
A 1 dimensional foliation which is Riemannian foliation with respect to no Riemannian metric
What is an example of a non vanishing smooth vector field on a manifold $M$ whose corresponding foliation is a Riemannian foliation with respect to no Riemannian metric on $M$
- 356
0
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0
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161
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A closed leaf with two different index with respect to two different Riemannian metrics
Inspired by this question about Jacobi equation. conjugate points and limit cycle theory we ask the following question:
Is there a geodesible 1 dimensional foliation $\mathcal{F}$ on a manifold $M$, ...
- 356
3
votes
0
answers
210
views
Jacobi equation and conjugate points on solution curves of the Van der Pol vector field
Let $X$ be a geodesible non vanishing vector field on a manifold $M$. Namely there is a Riemannian structure $(M,g)$ such that all integral curves of $M$ are unparametrized geodesics of the ...
- 356
3
votes
0
answers
145
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Naturality of geodesic flow
Let $\texttt{Man}$ be the category of smooth manifolds with local diffeomorphisms as morphisms, and $ \texttt{Bun}$ --- the category of bundles (affine bundles or just fibre bundles, if necessary) and ...
- 2,934
1
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0
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117
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Relation between the distance projective maps and their angles
Let $f:N \to \mathbb{R}^2$
be a differentiable map of smooth manifolds. Let $\mathbb{R}^2$ be decomposed as a direct sum of line bundles, i.e. $\mathbb{R}^2=E(x) \oplus F(x)$, where $F(x)$ and $E(x)$ ...
- 1,043
3
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0
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187
views
Exponential map of cotangent bundle and Morse theory on based loop space
Consider $M$ to be a compact manifold and $q_0,q_1\in M$. Let $L_t$ be a lagrangian and $\mathcal{E}_L$ the lagrangian action functional on the based loop space $\Omega(M,q_0,q_1)$ defined as $\...
- 791
5
votes
1
answer
228
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An example of an SRB measure which is not a physical measure
Let $f:M \to M$ be a $C^{1}$ diffeomorphism on a compact Riemannian manifold with a normalized Riemannian volume $\mathrm{Leb}$. Given an $f$-invariant Borel probability $\mu$ in $M$, we call the ...
- 1,043
2
votes
0
answers
71
views
Domain of definition of a certain mapping
Suppose we have a compact smooth riemannian manifold $(M,g)$ and a $\mathcal{C}^2$ diffeomorphism $f$ of this manifold.
I am studying the mapping
$$\tilde{f}(x)= \exp_{f(x)}^{-1} \circ f \circ \exp_x \...
3
votes
0
answers
134
views
Is there a non geodesible vector field $P\partial_x+Q\partial_y$ which satisfies $P_xP_y+Q_xQ_y=0$
Inspired by the following two posts
Finding a 1-form adapted to a smooth flow
Does $P_xP_y+Q_xQ_y=0 \implies$ "non-existence of limit cycle" for $P\partial_x+Q\partial_y$"? (Complex ...
- 356
5
votes
2
answers
433
views
On which closed Riemannian manifolds are geodesics always recurrent?
Let $M$ be a closed Riemannian manifold. What are the necessary and sufficient conditions on $M$ to ensure that for every point $p \in M$, and every geodesic $\gamma: [0, \infty) \to M$ with $\gamma(0)...
- 6,155
1
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0
answers
61
views
Is existence of a limit cycles an obstruction for a vector field to be a global Jacobi field?
Is there a Riemannian metric on $S^2$ and a vector field $X$ on $S^2$ with the following two properties?
The vector field $X$ is globaly a Jacobi field in the sense that for every point $x\in S^2$ ...
- 356
5
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0
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149
views
Are all linear vector fields geodesible vector fields?
I had already asked this question in MSE then I ask here at MO.
Assume that $A\in M_n(\mathbb{R})$ is a non singular matrix.
Is the flow of the linear vector field $X'=AX$ a geodesible flow on $\...
- 356
1
vote
0
answers
85
views
Dynamical obstructions for a vector field $X$ whose adjoint operator $ad_X$ sends a global orthonormal frame to a set of mutually orthogonal vectors
Let $X$ be a vector field on a parallelizable manifold $M$.
Can we equipe $M$ with a Riemannian metric such that we have at least one global orthonormal frame $\{V_1,V_2,\ldots,V_n \} $ such that $[...
- 356
5
votes
0
answers
104
views
Dynamical obstructions for a vector field whose derivation sends an orthonormal set to a mutually Sasakian orthogonal vectors
We ask two related questions which are inspired by this MO question Does $P_xP_y+Q_xQ_y=0 \implies$ "NONEXISTENCE OF LIMIT CYCLE for $P\partial_x+Q\partial_y$"? (Complex Dilatation and Limit ...
- 356
3
votes
0
answers
73
views
The diversity of Riemannian metrics adapted to a given foliation، A Krein Millman view point(2)
Inspired by this answer to the linked question we add a more bounded conditions to this post. This question is asked seperately because the previous one had a complete answer so we did not revise ...
- 356
5
votes
1
answer
204
views
The diversity of Riemannian metrics adapted to a given (1 dimensional) foliation, A Krein Millman view point
Let $X$ be a Kronecker vector field on the two dimensional torus $\mathbb{T}^2$. Let $K$ be the space of all 1- forms $\alpha$ of class $C^1$ on $\mathbb{T}^2$ which satisfy $d\alpha=0,\;\alpha(X)=1$...
- 356
5
votes
1
answer
597
views
A vector field whose flow has constant singular values
$\newcommand{\tr}{\operatorname{tr}}$
$\renewcommand{\div}{\operatorname{div}}$
Let $D \subseteq \mathbb{R}^2$ be the closed unit disk. Given a vector field $X$ on $D$, let $\psi_t$ be its flow.
Does ...
- 6,741
2
votes
0
answers
479
views
A Fourier elliptic vector field on a Riemannian manifold
Motivation for this question:
Let $X$ be a vector field on a manifold $M$. Obviously the differential operator $D:C^{\infty}(M)\to C^{\infty}(M)$ with $D(f)=X.f$ is not an elliptic opetator when $\...
- 356
5
votes
1
answer
164
views
A non-geodesible foliation of $S^3$ or $S^2\times S^1$
Is there a $1$-dimensional foliation of $S^3$ which is not a geodesible foliation? Is there a $1$-dimensional foliation of $S^2\times S^1$ which is not a geodesible foliation?
If the answer is ...
- 356
3
votes
0
answers
160
views
Non-existence of a parametrically compatible metric to a complete geodesible vector field on $\mathbb{R}^2\setminus\{p,q\}$
Inspired by this answer to the question entitled "Possible isometry groups of open manifolds" we ask the following question:
Is there a complete vector field $X$ on $\mathbb{R}^2\...
- 356
3
votes
0
answers
151
views
Anosov flows on non-compact manifold
So, when defining Anosov flows on a compact manifold specifying the Riemannian metric is not necessary as any two are equivalent. So, my question is:
Given a non compact manifold $M$ and a proper ...
- 229
7
votes
1
answer
258
views
Does Anosov geodesic flow imply asphericity?
Let $(M, g)$ be a closed smooth Riemannian manifolds with Anosov geodesic flow, does it implies that $M$ is an aspherical manifold?
I am thinking it is not known yet?
- 481
6
votes
2
answers
624
views
On equation $\Delta \circ \partial/\partial X=\partial/\partial X \circ \Delta$ on a Riemannian manifold
Assume that $M$ is a compact Riemannian manifold whose Laplacian is denoted by $\Delta$. Assume that the Euler characteristic of $M$ is zero. Does $M$ admit a non vanishing vector field ...
- 356
2
votes
0
answers
150
views
Global solution of second order ODE defined on riemannian manifold
Consider the differential equation $\nabla \dot X + \frac{3}{t} \dot X + gradf(X) =0$, defined on a riemannian manifold $(M,g)$ ($ \nabla$ is the Levi-Civita connection and $gradf(X)$ is the ...
- 335
6
votes
0
answers
283
views
A concept weaker than geodesibility of flows which is possibly useful in limit cycle theory
The main objective of this post is to apply the Gauss Bonnet Theorem to count the number of limit cycles of a polynomial vector field as described in this MO post and its linked MO posts But in this ...
- 356
7
votes
0
answers
452
views
Geometric bang-bang theorem for nonlinear optimal control
The classical bang-bang theorem is usually stated for linear systems (e.g. Control Theory from the Geometric Viewpoint by Agrachev-Sachkov, p. 209). Sussman proved a nice generalization for systems ...
- 1,731
1
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0
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218
views
Desingularization of the zero section of $TM$ as the manifold of singularities of the geodesic flow
However the method of "Blowing up of singularities" is initially introduced for an isolated singularity, but this method have been generalized to blowing up of a "Manifold of singularities"...
- 356
4
votes
2
answers
411
views
A kind of converse to the Hopf theorem on ergodicity of geodesic flow in negative curvature
Is there a 2 dimensional Riemannian manifold $M$ whose curvature is not negative but its geodesic flow is an ergodic flow?
- 356
1
vote
1
answer
217
views
Ergodicity of geodesic flow in negative curvatutre as a possible obstruction for consideration of limit cycles as closed geodesics(4)
Does the ergodicity of geodesic flow of compact surfaces with negative curvature stile hold for non compact case?
Is not the ergocity theorems of geodesic flow an obstruction to have a ...
- 356
3
votes
1
answer
191
views
Is there a connection $\nabla$ for which this particular non geodesible vector field $X$ satisfy $\nabla_X X=0$?
Let $X$ be the following vector field on $\mathbb{R}^2\setminus \{0\}$
\begin{align}
x' &= x\,(1-x^2-y^2)(x^2+y^2-3) - y\,(2-x^2-y^2)\\
y' &= y\,(1-x^2-y^2)(x^2+y^2-3) + x\,(2-x^2-y^2).
\...
- 356
5
votes
1
answer
680
views
Two questions on "foliation by geodesics"
I would appreciate if you consider the following two questions on $1$ dimensional foliations whose leaves are geodesic.
1)Assume that $M$ is a Riemannian manifold which is either an open ...
- 356
3
votes
0
answers
360
views
(Some possible obstructions to ) Limit cycles as closed geodesics(3)
First we explain our Motivation:
Motivation:
First note that there is no a Riemannian metric on an open set of the plane which possess two nested closed geodesics $\gamma_1, \gamma_2$ ...
- 356
8
votes
1
answer
365
views
Can a harmonic vector field possess a limit cycle?
Can a harmonic vector field $X$ on a Riemannian surface $(M,g)$ possess a limit cycle(An isolated periodic orbit)?
Note that the Laplacian of a vector field is defined via natural correspondence ...
- 356
2
votes
1
answer
181
views
Riemannian metric adapted to singular $1$-dimensional foliation
Is there a real analytic vector field $X$, locally defined around $0\in \mathbb{R}^{2n}$, with the following properties:
1) The origin is an isolated singularity for $X$ and its linear ...
- 356
7
votes
1
answer
841
views
Hilbert 16th problem via hyperbolic geometry
More than 16 years ago, I heard from someone that he thinks that there is a possible relation between Hilbert's 16th problem(for $n=2$) and Hyperbolic geometry. He says that a ...
- 356
3
votes
1
answer
195
views
An explicit formula for a flat metric compatible to certain polynomial vector field with center
Let $X$ be the following vector field on the plane:
$$\begin{cases} x'=y\\ y'=-x-x^3\end{cases}\;\;\;\;\;(X)$$
The vector field $ (X)$ has a non isochronous center at the origin.The ...
- 356
3
votes
1
answer
156
views
Is the space of harmonic functions invariant under the derivational operator associated with a geodesible flow?
Assume that $V$ is a vector field on a
Riemannian manifold $(M,g)$ with natural volume form $\Omega$ arising from $g$.
Assume that the solution curves of $V$ are parametrized geodesics of the ...
- 356
7
votes
0
answers
521
views
Limit cycles as closed geodesics(2)
Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the ...
- 356
2
votes
1
answer
467
views
A curvature description for center condition for quadratic vector field
We consider the quadratic vector field $V$ $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y)
\end {cases}\;\;\;\;(V)$$
where $P,Q \in \mathbb{R}[x,y]$ are polynomials of degree $2$ with $P(0,0)=Q(0,0)=...
- 356
3
votes
0
answers
165
views
Is a non vanishing holomorphic vector field necessarily a geodesible vector field?
Motivated by the "The obvious Fact" part of this answer,, we ask the following question:
First we recall a definition, which is used in the above link:
Definition: A non vanishing vector ...
- 356
3
votes
0
answers
165
views
Flat Riemannian metrics adapted to quadratic vector fields with center
Assume that $P(x,y),Q(x,y)\in \mathbb{R}[x,y]$ are two polynomials of degree $2$ with $P(0,0)=Q(0,0)=0.$
Suppose that the vector field $$\begin{cases} x'=P(x,y)\\ y'=Q(x,y) \end{cases}$$ has a center ...
- 356
5
votes
1
answer
208
views
Non conformally geodesible vector field
What is an example of a smooth vector field $V$ on an open set of the plane which is a geodesible vector field but there is no a conformal metric $g$ such that $V$ is geodesible ...
- 356
4
votes
0
answers
495
views
Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)
This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow.
Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
- 356
4
votes
1
answer
735
views
A certain generalization of the Poincare Bendixson theorem
Assume that we have a $n-1$ dimensional integrable distribution $D $ on $\mathbb{R}^n \setminus \{0\}$ which generates a foliation $\mathcal{F}$. We fix an orientation for $D$.(For $n=2$ we ...
- 356
44
votes
5
answers
6k
views
Finding a 1-form adapted to a smooth flow
Let $M$ be a smooth compact manifold, and let $X$ be a smooth vector field of $M$ that is nowhere vanishing, thus one can think of the pair $(M,X)$ as a smooth flow with no fixed points. Let us say ...
- 114k
1
vote
0
answers
308
views
A differential operator associated with a vector field on the torus
Assume that $X$ is a non vanishing vector field on the torus $\mathbb{T}^2$.
We define two linear operators $T,S$ on the space of smooth functions on $\mathbb{T}^2$ as follows:
$T(f)=...
- 356
3
votes
1
answer
251
views
A non vanishing vector field compatible to a Riemannian metric
Assume that $(M, g)$ is a connected Riemannian manifold which is either open or is compact with zero Euler characteristic.
Is there a non vanishing vector field $X$ on $M$ such ...
- 356