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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 ...
Terry Tao's user avatar
  • 114k
12 votes
3 answers
2k views

Limit cycles as closed geodesics (in negatively or positively curved space)

Updated 1/25/2023 I just added a related post below: Jacobi fields, Conjugate points and limit cycle theory EDIT: Here is a related post which concern quadratic vector fields rather than Van ...
Ali Taghavi's user avatar
73 votes
10 answers
11k views

Riemannian surfaces with an explicit distance function?

I'm looking for explicit examples of Riemannian surfaces (two-dimensional Riemannian manifolds $(M,g)$) for which the distance function d(x,y) can be given explicitly in terms of local coordinates of ...
Terry Tao's user avatar
  • 114k
46 votes
2 answers
11k views

Does the curvature determine the metric?

I ask myself, whether the curvature determines the metric. Concretely: Given a compact manifold $M$, are there two metrics $g_1$ and $g_2$, which are not everywhere flat, such that they are not ...
Bernhard Boehmler's user avatar
14 votes
2 answers
871 views

Are all maps $\mathbb{R}^2 \to \mathbb{R}^2$ with fixed singular values affine?

Let $f:\mathbb{R}^2 \to \mathbb{R}^2$ be a smooth map whose differential has fixed distinct singular values $0<\sigma_1<\sigma_2$ and an everywhere positive determinant (which is the product $\...
Asaf Shachar's user avatar
  • 6,741
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 ...
Ali Taghavi's user avatar
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)=...
Ali Taghavi's user avatar
36 votes
10 answers
6k views

Determining a surface in $\mathbb{R}^3$ by its Gaussian curvature

A curve in the plane is determined, up to orientation-preserving Euclidean motions, by its curvature function, $\kappa(s)$. Here is one of my favorite examples, from Alfred Gray's book, Modern ...
Joseph O'Rourke's user avatar
19 votes
4 answers
3k views

Equations satisfied by the Riemann curvature tensor

It is well known that the Riemann curvature tensor of a metric satisfies \begin{eqnarray} R_{jikl}=-R_{ijkl}=R_{ijlk},(1)\\ R_{klij}=R_{ijkl},(2)\\ R_{i[jkl]}=0 \mbox{(1st Bianchi identity)}.(3) \end{...
asv's user avatar
  • 21.8k
10 votes
2 answers
926 views

Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?

This is a cross-post. While working on a variational problem, I have reached to the following question. Let $0<\sigma_1<\sigma_2$ satisfy $\sigma_1\sigma_2=1$, and let $D \subseteq \mathbb{R}^2$...
Asaf Shachar's user avatar
  • 6,741
7 votes
1 answer
898 views

Sharp Gaussian upper bounds on Heat Kernel

I am looking for references (with proof) for the following statement: Let $(M, g)$ be a Riemannian manifold with bounded curvature and let $p_t(x , y)$ be the heat kernel of $M$. Let $K$ be compact ...
Matthias Ludewig's user avatar
110 votes
6 answers
16k views

When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?

As we all know, for a Riemannian manifold $(M,g)$, there exists a unique torsion-free connection $\nabla_g$, the Levi-Civita connection, that is compatible with the metric. I was wondering if one can ...
Jean Delinez's user avatar
  • 3,399
17 votes
2 answers
5k views

Square of the distance function on a Riemannian manifold

Let $(M^n,g)$ be a smooth Riemannian manifold. Consider the square of the distance function $$dist^2\colon M\times M\to \mathbb{R}$$ given by $(x,y)\mapsto dist^2(x,y)$. It is easy to see that this ...
asv's user avatar
  • 21.8k
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(...
Ali Taghavi's user avatar
43 votes
5 answers
8k views

A geometric interpretation of the Levi-Civita connection?

Let $M$ be a Riemannian manifold. There exists a unique torsion-free connection in the (co)tangent bundle of $M$ such that the metric of $M$ is covariantly constant. This connection is called the Levi-...
algori's user avatar
  • 23.5k
24 votes
4 answers
3k views

Obtain Lorentzian manifolds from Riemannian ones by Wick rotation

In some cases, Wick rotation of a metric, formally consisting in substituting a coordinate with i times the coordinate itself, allows one to construct a Riemannian manifold starting from a Lorentzian ...
GFR's user avatar
  • 639
19 votes
2 answers
2k views

Does a Kähler manifold always admit a complete Kähler metric?

Every smooth manifold admits a complete Riemannian metric. In fact, every Riemannian metric is conformal to a complete Riemannian metric, see this note. What about in the Kähler case? Does a Kähler ...
Michael Albanese's user avatar
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, ...
Bence Racskó's user avatar
14 votes
4 answers
6k views

When is a Riemannian metric equivalent to the flat metric on $\mathbb R^n$?

I'm looking for an easily-checked, local condition on an $n$-dimensional Riemannian manifold to determine whether small neighborhoods are isometric to neighborhoods in $\mathbb R^n$. For example, for ...
Theo Johnson-Freyd's user avatar
9 votes
1 answer
589 views

Injectivity radius of the Sasaki metric

Let $(Q,g)$ be a (compact) Riemannian manifold with injectivity radius $\rho>0$. There is a natural metric $\tilde g$ on the tangent bundle $TQ$ which is known as the Sasaki metric and which makes $...
Dawidow's user avatar
  • 125
8 votes
1 answer
599 views

Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have distinct singular values?

$\newcommand{\SO}[1]{\text{SO}(#1)}$ $\newcommand{\dist}{\operatorname{dist}}$ Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth. Set $$...
Asaf Shachar's user avatar
  • 6,741
7 votes
4 answers
2k views

Do cotangent bundles have "bounded geometry"?

I have often heard the phrase "a manifold $M$ has bounded geometry" thrown around without ever seeing a precise definition of what this means. Apparent examples are compact manifolds and $\mathbb{R}^n$...
ss78's user avatar
  • 71
6 votes
1 answer
475 views

Differentiability of distance function at first conjugate point

Formulation of question: Consider a complete, simply connected Riemannian manifold $M$ with Riemannian metric $d$. For $x,y \in M$ that are distinct but close enough to each other, there is a geodesic ...
Chee's user avatar
  • 984
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 ...
Ali Taghavi's user avatar
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 ...
Ali Taghavi's user avatar
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 $\gamma_1, \gamma_2$ ...
Ali Taghavi's user avatar
43 votes
11 answers
15k views

Open questions in Riemannian geometry

What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.
32 votes
4 answers
4k views

Largest hyperbolic disk embeddable in Euclidean 3-space?

Hilbert proved that there's no complete regular ($C^k$ for sufficiently large $k$) isometric embedding of the hyperbolic plane into $\mathbb{R}^3$. On the other hand, the pseudosphere is locally ...
j.c.'s user avatar
  • 13.6k
24 votes
5 answers
6k views

Curvature and Parallel Transport

Here is an updated formulation of the question, which is more precise and I think completely correct: Suppose $M$ is a Riemannian manifold. Pick a point $p$ in $M$ and let $U$ be a neighborhood of ...
Paul Siegel's user avatar
  • 29.2k
22 votes
4 answers
3k views

What is the analog of the "Fundamental Theorem of Space Curves," for surfaces, and beyond?

The "Fundamental Theorem of Space Curves" (Wikipedia link; MathWorld link) states that there is a unique (up to congruence) curve in space that simultaneously realizes given continuous curvature $\...
Joseph O'Rourke's user avatar
20 votes
5 answers
2k views

Smoothness of the closest point on a submanifold

Let $(M,g)$ be a smooth Riemannian manifold, and let $S \subseteq M$ be a compact submanifold. Assume that for each $p \in M$, there exist a unique closest point on $S$, i.e a unique point $\tilde s(...
Asaf Shachar's user avatar
  • 6,741
20 votes
3 answers
9k views

Curvature of a Lie group

Since a lie group is a manifold with the structure of a continuous group, then each point of the manifold [Edit: provided we fix a metric, for example an invariant or bi-invariant one] has some scalar ...
Matt's user avatar
  • 251
20 votes
4 answers
3k views

Can a manifold have a curvature-free connection that is not torsion-free?

Suppose I have a smooth manifold with a tangent bundle, and I have a connection. If this connection is curvature-free, is it guaranteed to be torsion-free? (I am not assuming a metric, just a finite-...
pavpanchekha's user avatar
  • 1,491
17 votes
4 answers
3k views

Green's operator of elliptic differential operator

Let $P:\Gamma(E)\rightarrow\Gamma(F)$ be an elliptic partial differential operator, with index $=0$ and closed image of codimension $=1$, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
mdg's user avatar
  • 376
15 votes
4 answers
918 views

Minimizing geodesics in incomplete Riemannian manifolds

Let $(M, g)$ be a Riemannian manifold, not necessarily complete. Let $x$ be a point in $M$, and let $r>0$ be such that the exponential map $\operatorname{exp}_x$ is defined on an open ball $B=B(0,r)...
macbeth's user avatar
  • 3,212
15 votes
3 answers
2k views

Characterizing Hessians among symmetric bilinear tensors

I apologize in advance if this is somewhat elementary, but: Let $(M,g)$ be a compact Riemannian manifold. Is there a "characterization" of which symmetric bilinear tensors $B\in Sym^2(M)$ ...
Renato G. Bettiol's user avatar
14 votes
1 answer
3k views

Is the cotangent bundle to a Kahler manifold hyperkahler?

Let me be more specific. Let $M$ be a Kahler manifold with Riemannian metric $g$ and complex structure $I$. Then $T^\ast M$ will also be Kahler with metric and complex structure induced from $M$ (I ...
Sam Gunningham's user avatar
12 votes
1 answer
3k views

how to define the injectivity radius of manifolds with boundary?

For manifolds without boundary one defines the injectivity radius as the maximal radius where the exponential map is a diffeomorphism. One can then show that the injectivity radius is the maximum ...
quarague's user avatar
  • 687
12 votes
1 answer
2k views

Isometry group of pseudo Riemannian manifold always a Lie group? (Myers-Steenrod)

Myers-Steenrod states that the isometry group of a Riemannian manifold is a Lie group. Is that also true for pseudo Riemannian manifolds? I didn't find anything related to that. Cheers
Stephan Meier's user avatar
11 votes
2 answers
2k views

Retraction of a Riemannian manifold with boundary to its cut locus

This question is edited following the comment of Joseph. He pointed out that the main object of the first version of this question is the cut locus. Recall that the cut locus of a set $S$ in a ...
Dmitri Panov's user avatar
  • 28.9k
10 votes
3 answers
757 views

Spin-H structures

Let us define a Spin-H structure as a reduction of a SO(n)-bundle by the group: $$Spin^H (n)=Spin(n) \times SU(2)/\{ 1,-1\}$$ The Spin-H structures are analogous to the well-known Spin-C structures ...
A.Balan's user avatar
  • 187
8 votes
3 answers
2k views

What does it mean that the Hessian is proportional to the metric?

Let $(M,g)$ be a smooth manifold equipped with a metric tensor $g$, and $f\in C^\infty(M)$ a regular function (i.e., with nowhere vanishing differential). Denote by $\mathrm{Hess}_g(f):=\nabla df$ ...
Giovanni Moreno's user avatar
8 votes
3 answers
419 views

Extension of a vector field to an orthonormal frame for a flat metric

Assume that $U$ is an open set in the plane and $X$ is a non vanishing vector field on $U$. Is there a non vanishing vector field $Y$ on $U$ such that the pair $\{X,Y \}$ plays the role of an ...
Ali Taghavi's user avatar
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 ...
Ali Taghavi's user avatar
6 votes
2 answers
1k views

Vector Fields in a Riemannian Manifold

Suppose $(M,g)$ is a Riemannian manifold. Is there a way to classify manifolds where there exists a vector field that commutes with the laplace beltrami operator? Thanks
Ali's user avatar
  • 4,135
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 ...
Ali Taghavi's user avatar
5 votes
2 answers
1k views

Continuity of Hausdorff measure on level sets

Let $\Omega\subset\mathbb{R}^2$ a open and bounded set with smooth boundary and $\phi:\Omega\to\mathbb{R}$ a smooth function such that: $\bullet$ $\phi^{-1}(0)\neq\emptyset$; $\bullet$ $\nabla\phi(x)\...
Bogdan's user avatar
  • 1,759
5 votes
2 answers
909 views

A Converse to the Gauss Bonnet Theorem

Let $S$ be a compact surface in $\mathbb{R}^{3}$ with the gauss normal map $N:S\to \mathbb{S}^{2}$. Assme that $\phi;\mathbb{S}^{2}\to S$ is a diffeomorphism. Put $F=N\circ \phi$ and represent $F:\...
Ali Taghavi's user avatar
5 votes
2 answers
480 views

Maximum symmetry metric on $ \mathbb{C}P^n $

Let $ M $ be a compact connected manifold. The degree of symmetry of $ M $, denoted $ N(M) $, is the maximum of the dimensions of the isometry groups of all possible Riemannian structures on $ M $. ...
Ian Gershon Teixeira's user avatar
4 votes
1 answer
1k views

Curvature as metric invariant

This is quite well-known: the ONLY metric invariants are curvature, its higher derivatives, and any possible contractions between them. The meaning of an invariant is, to put it simply, a tensor ...
Caramba's user avatar
  • 133

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