# Tagged Questions

**1**

vote

**1**answer

231 views

### Symmetry on a sphere

Let $u$ be a smooth function on the sphere $S^2$. Suppose there exists $C>0$ such that for all $R \in SO(3)$, the area of every connected component of $\{x\in S^2: u(x)> u(Rx)\}$ is at least ...

**3**

votes

**1**answer

194 views

### Exponential mapping versus flow

In Hamilton's article on the Nash-Moser Theorem, he gives the map that maps a vector field $X$ to its flow $e^{tX}$ in $\mathrm{Diff}(M)$ as an example where the implicit function theorem in Frechet ...

**2**

votes

**1**answer

134 views

### Parameter dependent differential equation in a Lie group

It is well-known that a linear differential equation in a finite-dimensional vector space depends continuously on some external parameters (for details see below). I search for an explicit reference ...

**4**

votes

**1**answer

262 views

### Pseudo-differential operators with compactly supported symbols

If the symbol $p(x,\xi)$ of a pseudodifferential operator $P$ has compact $x$-support, then for any Schwartz function $f$, $Pf$ has compact $x$-support.
Is the reverse true? Namely that if some PDO ...

**0**

votes

**1**answer

104 views

### Application of the inverse theorem function on singular points

This question is related to this one A question about the inverse theorem function in $\mathbb{R}^n$. In the response I received, I was told that the issue was related to the singularities, as I do ...

**0**

votes

**2**answers

209 views

### A question about the inverse theorem function in $\mathbb{R}^n$

Let $f:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be a continuously differentiable mapping. We assume that the set $$\{x\in\mathbb{R}^{n};j(f)(x)=0\}$$ is a hypersurface of $\mathbb{R}^{n}$, where $j(f)(x)$ ...

**1**

vote

**0**answers

80 views

### Problem reduced to analyzing solutions of a family of nonlinear systems of equations

I was able to reduce a research problem relating to normal numbers to analyzing the solutions of the following family of nonlinear systems of equations:
\begin{align*}
c_0+c_1+\cdots+c_{t-1}&=2t\\
...

**2**

votes

**1**answer

370 views

### Existence of Geodesics in continuous metrics

I learned that if we are given a $C^0$ Riemannian metric on a smooth manifold $M$, geodesics (i.e. length minimizing curves) are absolutely continuous, and if the metrics is $C^{0,\alpha}$, then the ...

**3**

votes

**0**answers

173 views

### Constructing a Sobolev space containing the differential k-forms of a Riemannian manifold

I am currently writing a paper about the Hodge theorem for an algebraic topology course. The specific formulation I am proving can be stated thus. Let $M$ be a compact, orientable n-dimensional ...

**14**

votes

**1**answer

416 views

### Counterexample to Sard Theorem for a not-C1 map

Is there a function $f: \mathbb{R} \rightarrow \mathbb{R}$ which is differentiable but not $C^{1}$, such that the image of the points where $f'(x) = 0$ has measure bigger than 0?
If the answer is no, ...

**20**

votes

**1**answer

744 views

### Are the only local minima of $\angle(v, Av)$ the eigenvectors?

Let $A$ be an invertible $n \times n$ complex matrix. For $v \in \mathbb{CP}^{n-1}$, define
$$d(v) = \frac{|\langle A \tilde{v}, \tilde{v} \rangle |^2}{ \langle A \tilde{v}, A \tilde{v} \rangle ...

**13**

votes

**1**answer

334 views

### Nonnegative smooth function as sum of squares of smooth functions

There is a famous open problem, whose solution is attributed to Paul Cohen, but no published paper seems to be available:
There exists $f\in C^\infty(\mathbb R,\mathbb R_+)$ such that $f$ is not a ...

**1**

vote

**1**answer

445 views

### Solutions to Heat Equations with Obstacles!

Consider a closed Riemannian manifold $(M,g)$ and a positive function $\psi: M \to R$. Fix a point $p \in M$, I have been struggling to construct a solution to the heat equation, $\partial_t u = ...

**2**

votes

**3**answers

407 views

### How do we use an Ehresmann connection to define a semispray?

Let $M$ be a differentiable manifold, let $TM$ be its tangent bundle, and consider $TTM$, the double tangent bundle.
Let $V \subseteq TTM$ denote the vertical subbundle, which is determined in a ...

**7**

votes

**1**answer

203 views

### Extremals versus minima for variational problems

A geodesic on a Riemannian manifold is generally not the shortest among nearby curves with the same endoints. But it can always be divided into parts each the shortest among nearby curves between its ...

**2**

votes

**0**answers

123 views

### Extrapolation on the p-norm sphere using the exponential map

Hi,
In order to follow a branch of solutions to an implicit equation on the manifold $M = \lbrace x \in \mathbb R^n, \|x\|_p = 1 \rbrace $, I'm interested in the following problem. Given two points ...

**6**

votes

**1**answer

348 views

### How “should” I define “absolutely continuous” functions on e.g. n-spheres?

(Am writing this post in a rush, out of office, so cannot give adequate links etc right now.)
There is a classical and well-understood definition of what it means for a continuous function ...

**6**

votes

**2**answers

577 views

### Linearly independent vector fields

Let $X_1,\dots,X_n$ be complete vector fields on $\mathbb R^n$ and suppose that $(X_1(p),\dots,X_n(p))$ is a basis for all $p \in \mathbb R^n$.
Question: Is it possible to choose a cube $C$ around ...

**4**

votes

**1**answer

583 views

### Results about existence/uniqueness of solution to Euler-Lagrange equations?

While studying calculus of variations, there is one question that I feel is missing in the texts I'm reading:
What can we say about the existence and/or uniqueness of solutions to Euler-Lagrange ...

**-1**

votes

**1**answer

278 views

### Global invertibility

A differentiable transformation of R^n at each point has an invertible derivative. Does it imply that the transformation is a global diffeomorphism?

**6**

votes

**1**answer

523 views

### On a compact manifold, what kind of function can be the Jacobian of a diffeomorphism?

I could not answer or find references of this question, even for the following special case:
On $S^2$ (the two-sphere equiped with the standard Riemannian metric), is every positive smooth function ...

**0**

votes

**0**answers

606 views

### First order differential equation of vector fields

Given vector fields $Y$, $Z$ on a (possibly compact) manifold $M$, I would like to know about the existence of solutions $X$ to the differential equation
$$ \nabla_Y X + a \cdot \mathrm{div}(Y)\cdot X ...

**4**

votes

**2**answers

600 views

### gradient of convex functions

Hello. Can somebody help me with the following question that I have thought over for quite some time, to no avail?
Let $f$ be a smooth function (class $\mathrm{C}^{\infty}$), $f:\mathbb{R}^n ...

**3**

votes

**1**answer

333 views

### Regarding Discrete Eigenvalues

For many eigenvalue problems for differential operators (for example the quantum harmonic oscillator (HO)), unless we impose some behaviour at infinity, the eigenvalues will not be discrete.
But, ...

**8**

votes

**1**answer

469 views

### A differential inequality needed to prove a theorem about odd-dimensional souls

I need a solution to this problem (which is really a calculus problem) in order to prove a rigidity result for open nonnegatively curved manifolds with odd-dimensional souls:
Suppose that ...

**1**

vote

**0**answers

214 views

### glue together a sequence of holomorphic forms

hallo,
my problem is the following: i have a finite sequence of holomorphic $k-$forms $\alpha_{k}$, each defined on open subsets $U_{k} \subset M$, where $M$ is a complex $n$-dimensional manifold, ...

**2**

votes

**1**answer

269 views

### harmonic function on surface

Dear all,
I am looking for reference books about real-valued harmonic functions on complete Riemannian surfaces, do you have any reference in your mind about this?
I found some books about harmonic ...

**25**

votes

**6**answers

5k views

### Why are there so many smooth functions?

I do understand that my question might seem a little bit ignorant, but I thought about it a lot and still can't wrap my head around it.
Analycity imposes very strong conditions on a map, from ...

**0**

votes

**0**answers

182 views

### Graph Laplacian to Continuous version

Has there been a study of vector laplacian that is a continuous version of a graph laplacian? Is there a good introduction to the topic?

**4**

votes

**1**answer

625 views

### Is there a smooth map f:R^n --> R^{n-1} whose image separates the rationals and others?

Let me make the question more precise:
Let A be the set of all points in R^n such that each of its coordinates is rational, and let B=R^n-A. My question is, is there a smooth map f:R^n --> R^{n-1} ...

**5**

votes

**0**answers

256 views

### Differential forms on the simplex which are “constant towards the boundary”

Let $\Delta^k$ the standard $k$-dimensional simplex, $\Delta^k=\{(x^0,\dots,x^k)\in \mathbb{R}^{k+1}|\sum_{i=0}^kx^i=1\}$, and let $\Omega^\bullet(\Delta^k)$ be the de Rham complex of smooth ...

**3**

votes

**1**answer

597 views

### diffeomorphic, holomorphic, biholomorphic

Let $\varphi : U \rightarrow X$ be a holomorphic mapping of some open set $U\subseteq\mathbb{C}$ into a complex $n-$dimensional manifold $X$. If we know that this mapping is diffeomorphic onto its ...

**0**

votes

**1**answer

292 views

### convergence of metrics

Hi, I have the following question: take a Riemannian manifold M, with a family of smooth metrics $g(t)$ in $[0,T)$, call $D_0$ the Levi-Civita connection of $g(0)$ and assume that for every $m\geq 0$
...

**4**

votes

**0**answers

129 views

### Question about density of $C^{\infty}(M,N)$ in $W^{1,p}(M,N)$ with $N$ not compact

Hi!
Let $M$ be a compact manifold possibly with boundary with $\dim(M)=m$, let $N$ be a non compact manifold with $\dim(N)=n$. Let me recall the definition of the sobolev space $W^{1,p}(M,N)$. ...

**4**

votes

**3**answers

1k views

### Analytical solutions of a differential equation (from Archimedes' Spiral)

There is a differential equation in polar coordinates:
$r'^2+r^2=(kt)^2$, $r(t=0)=0$, k- Const.
I've found that
a) if $\phi \in (0,t)$, t is quite small, then $r(\phi) \approx k/2 *\phi^2$
b) if ...

**6**

votes

**3**answers

627 views

### Hidden convexity

Suppose you are given a domain $\Omega \subset \mathbb{R}^n,$ and a (Morse) function $f: \Omega \rightarrow \mathbb{R},$ all of whose critical points are positive-definite.
The question is: is there ...

**0**

votes

**0**answers

387 views

### Analogue of Whitney's extension theorem

So let $n,m$ be two strictly positive numbers. Let $A\subseteq\mathbf{R}^n$ be
a compact $C^{\infty}$-submanifold (possibly with boundary and of real dimension not necessarily equal to $n$). Let ...

**6**

votes

**1**answer

321 views

### Smooth and analytic structures on low dimensional euclidian spaces

So it is relatively easy to show that there exists only one smooth structure on
the real line $\mathbb{R}$. So here are 2 natural questions:
Q1: Up to equivalence, is there only one real analytic ...

**4**

votes

**1**answer

610 views

### Solution of Plateau Problem for a simple, smooth closed curve on a Riemannian Manifold (Kahler) gives a surface that can be parametrized by a closed disk?

Hi,
Perhaps it's a stupid question, in that case i'll delete it.
Let M be a compact orientable smooth (Kahler if changes things) manifold of dimension $dim_{\mathbb{R}}(M)=2n$ with $n\geq1$, let ...

**0**

votes

**1**answer

276 views

### question on the proof of density of $C^{\infty}(M,N)$ in the sobolev space $W^{1,m}(M,N)$

Hi,
i'm reading the proof of the fact that $C^{\infty}(M,N)$ is dense in the sobolev space $W^{1,m}(M,N)$, where $M,N$ are compact riemannian manifolds of dimension respectively $m,n$.
I recall ...

**1**

vote

**1**answer

201 views

### A question about to computing a integration

Let $M$ be a smooth manifold of 2n-dim, $v$ be a map from $M$ to the matrix of order $m\times m$.
We call $p\in M$ is the singularity, if $v(p)$ is non-invertible. Suppose $v$ is smooth and the ...

**2**

votes

**1**answer

553 views

### Reference for existence and uniqueness of differential equations for low differentiability?

My specific situation is that I have a non-spacelike continuous future directed curve $\gamma:[0,a)\to M$ in a Lorentzian manifold. The curve must necessarily satisfy a local Lipschitz condition and ...

**5**

votes

**4**answers

1k views

### How does curvature change under perturbations of a Riemannian metric?

Let $M$ be a compact subset of $\mathbb R^2$ with smooth boundary, and let $g$ be a Riemannian metric on $M$. If $g'$ is another Riemannian metric which is "close" to $g$, then they should have ...

**11**

votes

**1**answer

541 views

### Reference for a particular Radon transform on non-positively curved spaces

Let me first recall that the classical Radon transform takes a (smooth compactly supported, say) function $f$ defined on $\mathbb{R}^n$ as an input, and gives as output the map $H\mapsto \int_H f$ for ...

**6**

votes

**1**answer

411 views

### Estimating the flow when we know the vector field

Suppose we have a $C^k$ vector field $v$ and let $\phi_t$ be the corresponding flow. I have estimates on $v$ and its derivatives: $|v| < C_0$, $|Dv| < C_1$, $|D^2v| < C_2$, ... $|D^kv| < ...

**22**

votes

**2**answers

2k views

### Examples of loss of regularity by “creation of topology”

I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem ...

**6**

votes

**2**answers

420 views

### Can I detect the point of impact without looking at it?

I'm going to postpone the motivation for this question because the question itself involves no complicated maths and may well have a very simple solution so I don't want to put anyone off with high ...

**83**

votes

**16**answers

11k views

### How do I make the conceptual transition from multivariable calculus to differential forms?

One way to define the algebra of differential forms $\Omega(M)$ on a smooth manifold $M$ (as explained by John Baez's week287) is as the exterior algebra of the dual of the module of derivations on ...

**21**

votes

**7**answers

2k views

### Rolle's theorem in n dimensions

This looks like a statement from a calculus textbook, which perhaps it should be.
"Rolle's theorem". Let $F\colon [a,b]\to\mathbb R^n$ be a continuous function such that F(a)=F(b) and F'(t) exists ...

**17**

votes

**5**answers

1k views

### Sheaves and Differential Equations

How do sheaves arise in studying solutions to ordinary differential equations?
EDIT: Is it possible to construct non-isomorphic sheaves on a domain $D \subset \mathbb{R}^n$ using solution sets to ...