# Tagged Questions

**3**

votes

**1**answer

201 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

**0**answers

96 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

**0**answers

77 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

**1**answer

168 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

**1**answer

284 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

**1**answer

203 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

**1**answer

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 ...

**3**

votes

**2**answers

350 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

**1**answer

220 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

**2**answers

330 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

**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 ...

**5**

votes

**1**answer

860 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

**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 ...