All Questions
10 questions
4
votes
1
answer
407
views
Lipschitz-regularity of partition of unity
Let $K$ be a compact subset of $\mathbb{R}^n$ and $\mathcal{U}$ be a finite collection of open subsets covering $K$ satisfying the minimality property: for every $U\in \mathcal{U}$, the sub-collection ...
- 195
5
votes
1
answer
207
views
The Lipschitz constant of convex sphere in $\mathbb{R}^3$
Is every convex sphere (in the sense of Alexandorff, which is the boundary of some convex body in $\mathbb{R}^3$) with Alexandorff curvature $\geq 1$, admitting a bijective map to the unit round ...
- 799
7
votes
1
answer
246
views
Currents in sub-Riemannian geometry
Federer and Fleming's notion of "currents" is well established so far, and starting from the seminal work of Ambrosio and Kirchheim, the notion of metric currents is well studied also. The ...
- 215
1
vote
0
answers
88
views
Density of $C^k$-functions with Lipschitz partial derivatives
Let $N$ and $M$ be complete Riemannian manifolds, of respective dimension $n$ and $m$ with $n,m\geq 1$. Let $C^{k,1}_b(N,M)$ be set of all bounded continuous functions $f:N\rightarrow M$ for which ...
- 5,405
5
votes
4
answers
589
views
Looking for a reference on conformal mapping on $\Bbb R^n$
A mapping $T: \Bbb R^n\to \Bbb R^n$ is said to be conformal if it is bijective and preserves angles, i.e.,
if $x, y: [0,1]\to \Bbb R^n$ are curves with $x(t_0)=y(t_0)$ then
$$\cos (Tx(t_0),Ty(t_0))= \...
- 1,101
4
votes
1
answer
224
views
When is the cut-locus normal coordinate collared
Let $(M,g)$ be a complete $d$-dimensional Riemannian manifold, $p \in M$ be fixed and let $C_p$ be the cut-locus of $p$.
Other than when $M$ is non-positively curved (in which $C_p=
\emptyset$ by ...
- 5,405
4
votes
3
answers
3k
views
Covariant derivative of determinant of the metric tensor
Let $(M,g)$ be a Riemannian manifold and $g$ the Riemannian metric in coordinates $g=g_{\alpha \beta}dx^{\alpha} \otimes dx^{\beta}$, where $x^{i}$ are local coordinates on $M$. Denote by $g^{\alpha \...
- 131
7
votes
2
answers
787
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 (...
- 13.5k
6
votes
1
answer
802
views
Approximation of a Sobolev function that has vanishing trace on the reduced boundary of a Caccioppoli (i.e. finite perimeter) set
For $\Omega\subset\mathbb{R}^N$ open and bounded, let $W^{1,p}(\Omega)$ denote the usual Sobolev space of $L^p(\Omega)$ functions with weak partial derivatives in $L^p(\Omega)$ and $W_0^{1,p}(\Omega)$ ...
- 143
28
votes
7
answers
5k
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 $F'(t)$ ...