All Questions
10 questions
3
votes
1
answer
189
views
Randomly perturbed function has no accumulated critical point almost surely?
Given a smooth function $f$ and a smooth manifold $\mathcal{M}$ in $\mathbb{R}^d$, define the set
$$
S(v):=\{x:{\rm Proj}_{T_x{\mathcal{M}}}(v)=\nabla_{\mathcal{M}}f(x)\}.
$$
Is correct to say that $S(...
- 43
11
votes
1
answer
451
views
Does every smooth map of rank at most d factor through a d-manifold?
Suppose $d≥0$, $m≥0$, $n≥0$, and $\def\R{{\bf R}} f\colon \R^m→\R^n$ is a smooth map
whose rank at any point of $\R^m$ is at most $d$.
Here and below, smooth means infinitely differentiable.
Can we ...
- 37.8k
2
votes
0
answers
354
views
Continuity of surface integrals on level sets
Let $\phi:\mathbb{R}^2\to\mathbb{R}$ such that $\phi^{-1}(0)\neq\emptyset$ and $\phi\in C^1(W)$ where $W$ is a compact neighborhood of $\phi^{-1}(0)$, with $\nabla\phi\neq 0$ in $W$. So there is some $...
- 1,759
3
votes
1
answer
97
views
Behaviour of mass for currents with disjoint supports
I am sorry if this is a basic question, but I don't think in MSE I will receive any answers.
Let $(M^3,g)$ be a compact and oriented Riemannian $3$-manifold. Let $\alpha$ and $\beta$ be the integral ...
- 2,095
6
votes
1
answer
1k
views
Fubini's theorem on arbitrary foliations
In what follows $ \mathbb{R}^{n+m} = \{(x,y): x \in \mathbb{R}^n, \ y \in \mathbb{R}^m \} \ .$
Suppose $G: U \to V $ is a $C^1$-diffeomorphism from an open subset of a manifold to an open subset of $...
- 622
0
votes
0
answers
96
views
If $M$ is a manifold, $x∈M$ and $d(x,ω)=\inf\{t>0:x+tω∈M\}$, does the pushforward of the solid angle measure under $S^2∋ω↦x+d(x,ω)ω$ admit a density?
Let $S^2$ denote the unit 2-sphere, $M$ be a 2-dimensional oriented embedded $C^1$-submanifold of $\mathbb R^3$ with $$d_M(x,\omega):=\inf\left\{t>0:x+t\omega\in M\right\}<\infty\;\;\;\text{for ...
- 167
2
votes
3
answers
803
views
A Curved/Warped Version of Fubini's Theorem
I will think of $ \mathbb{R}^{n+m}$ as $\mathbb{R}^n \times \mathbb{R}^m$.
Let $ V \subset \mathbb{R}^{n+m}$ be open and $g:V \to U \subset \mathbb{R}^{n+m} $ be a $C^1$ diffeomorphism. For a fixed ...
- 622
1
vote
0
answers
183
views
Total Mean Curvature as a integral on the whole space
It is well known from De Giorgi that we may express the surface area of a domain $\Omega\subset\mathbb{R}^N$ as:
$$
\int_{\partial\Omega} 1\ d\sigma=\int_{\Omega} ||\nabla H(\phi(x))||\ dx=\int_{\...
- 1,759
3
votes
0
answers
91
views
Asymptotical control of the measure of tubes covering subsets of fixed Hausdorff dimension
(A version of this question was posted on math stack exchange)
Let $M$ be a $C^1$ submanifold of dimension $n$ of $\mathbb{R}^N$, and denote $\mu$ the standard surface measure on $M$.
Consider a ...
- 1,035
3
votes
1
answer
938
views
Stokes theorem for manifolds with boundary as disjoint union of submanifolds
Looking at the generalizations of Stokes theorem, I did find a version for manifold with corners, but I was surprised this generalization doesn't contain a simple example such as the cone. So my ...
- 549