3
votes
0answers
57 views

“Integrating” solenoidal vector fields

Let $U$ be a bounded open subset of $\mathbb{R}^d$ with Lipschitz boundary, and $g \in L^2(U,\mathbb{R}^d)$ be a solenoidal vector field (i.e. $\nabla \cdot g = 0$). Then $g$ can be written in the ...
2
votes
1answer
193 views

The class of uniformly accelerated curves and surfaces

Once upon a time I was travelling by train and noticed an intresting optic effect I started to think about in terms of math. Let's consider two examples of curves: 1)The curve defined by the ...
1
vote
0answers
69 views

Structure of the zero set of analytic maps (Lojasiewicz’s Structure Theorem for Varieties)

The Lojasiewicz’s Structure Theorem for Varieties states that for $\Phi:\mathbb{R}^n\rightarrow \mathbb{R}$ real analytic with $\Phi(0)=0$ the set $\Phi^{-1}(0)$ is locally a union of subvarieties of ...
3
votes
0answers
142 views

Boundary of an open, bounded and convex set in $\mathbb{R} ^n$

Let $U$ be an open, bounded and convex set in $\mathbb{R} ^n$. Since $\partial U$ is a rectifiable set it follows that up to a set of $H^{n-1}$-measure zero $\partial U$ is contained in a countable ...
7
votes
2answers
334 views

Riemannian distance functions on the real line

A distance function $d: \mathbb{R} \times \mathbb{R} \rightarrow [0,\infty)$ that is defined by a smooth Riemannian metric on the real line satisfies the following properties: $d$ is a length metric ...
5
votes
3answers
493 views

Non-continuous differentiability for differential forms

Generally when working with differential forms, one assumes that they are continuously differentiable, i.e. $C^r$ for some $1\le r \le \infty$. Under this hypothesis, one can define the exterior ...
3
votes
1answer
211 views

Preimage of a smooth function

Suppose we are given a smooth function $f\colon \mathbb{R}^n \rightarrow \mathbb{R}$ and some number $c$. What can be said about the preimage $f^{-1}(c)$. There's the theorem on regular preimages, ...
1
vote
0answers
113 views

Extension of diffeomorphisms preserving bilateral bounds of the derivatives

Suppose $f$ is a $C^k (1\leq k\leq\infty)$ function from the unit ball $\mathcal{B}$ in $\mathbb{R}^n$ to itself, which is a diffeomorphism from the domain to its image, with the upper and lower ...
2
votes
0answers
85 views

convolution of surface measures

Thanks for reading my question. Assume we are given two compact (possibly with boundary) $(n-1)$-dimensional $C^{\infty}$ manifolds $M_1$ and $M_2$ embedded in $\mathbb{R}^n$, with induced measure ...
4
votes
1answer
171 views

Ideals in a subalgebra in $C^\infty(M)$

Let $M$ be a (usual, finite dimensional) smooth manifold, and $C^\infty(M)$ the algebra of (real valued) smooth functions on $M$. For each point $a\in M$ and for each subalgebra $A$ in $C^\infty(M)$ ...
0
votes
1answer
323 views

Global Implicit Function Theorem

Let $F:\mathbb R^2\rightarrow\mathbb R$ be a measurable function. Under what conditions on $F$ does there exist a function $\theta:\mathbb R^2\rightarrow\mathbb R$ such that $F(x,\theta(z,x))=z$ for ...
4
votes
1answer
210 views

Morse lemma with least amount of regularity.

I recently came across with $C^2$ Morse functions in my work and as I was reviewing some of the stuff I learned about Morse theory, I noticed that all the proofs of the Morse lemma I could come across ...
1
vote
1answer
303 views

Singular conformally-Euclidean metrics

Suppose $W : \Bbb{R}^n \to \Bbb{R}_+$ is a continuous, positive function, with exactly $n$ zeros $\alpha_1,...,\alpha_n$. Define the following 'distance': $$ d(\alpha_i,\alpha_j)=\inf\{\int_0^1 ...
4
votes
2answers
367 views

implicit function theorem for algebraic sets

We know by the standard Implicit Function Theorem that If $f:\mathbb R^4\rightarrow\mathbb > R^2$ is a polynomial (or in fact any continuously differentiable function), then there is a ...
1
vote
1answer
221 views

Conformal Extension from a closed set to open

Let $Q = \{(x,y): x,y\geq 0\} $ be the 1st quadrant of $\mathbb R^2$, and $f$ is a function defined on it such that all the partial derivative(any order) of $f$ exists and continuous. By Whitney ...
2
votes
2answers
346 views

Estimating the Hausdorff measure of a subset of the sphere

Let $f: S^{n-1}\to \mathbb{R}$ be a continuous function ($S^{n-1}\subset \mathbb{R}^n$ is the unit sphere), $f(a)>0$ and $f(b)<0$ for certain points $a,b\in S^{n-1}$. By continuity these ...
0
votes
0answers
418 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 ...
5
votes
1answer
901 views

Gluing two diffeomorphisms together

A fundamental construction in a first course on manifolds is to build a smooth function $\psi\colon \mathbb{R} \to \mathbb{R}$ with the property that for some $0<\delta<\epsilon$ we have ...
21
votes
7answers
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 ...