Search Results
| Search type | Search syntax |
|---|---|
| Tags | [tag] |
| Exact | "words here" |
| Author |
user:1234 user:me (yours) |
| Score |
score:3 (3+) score:0 (none) |
| Answers |
answers:3 (3+) answers:0 (none) isaccepted:yes hasaccepted:no inquestion:1234 |
| Views | views:250 |
| Code | code:"if (foo != bar)" |
| Sections |
title:apples body:"apples oranges" |
| URL | url:"*.example.com" |
| Saves | in:saves |
| Status |
closed:yes duplicate:no migrated:no wiki:no |
| Types |
is:question is:answer |
| Exclude |
-[tag] -apples |
| For more details on advanced search visit our help page | |
Results tagged with riemannian-geometry
Search options not deleted
user 171439
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
3
votes
1
answer
217
views
Local existence of flat metrics with degenerate singular values
It has been proved that,
If $\lambda_1,\,\lambda_2,\cdots,\lambda_n$ are real analytic functions from $\mathbb{R}^n$ to $\mathbb{R}$, such that $\lambda_i(0)\neq \lambda_j(0)$ for $i\neq j$, then ther …
- 134
10
votes
0
answers
188
views
Non-trivial $\mathbb{R^3}\rightarrow\mathbb{R^3}$ maps with constant singular values
It can be proved that all $\mathbb{R^2}\rightarrow\mathbb{R^2}$ mappings with constant singular values are affine. In three dimensions, however, there are non-trivial examples, like
$$
\begin{align}
…
- 134
2
votes
0
answers
118
views
Characterization of planar domains onto which a unit disk can be mapped with constant singul...
It can be shown that there are (smoothly bounded, Jordan) domains $E\subset \mathbb{R}^2$ which are $\textit{not}$ images of mappings $f$ from the unit disk (or any other planar domain), such that $\m …
- 134
4
votes
1
answer
276
views
Mappings between 2-manifolds with symmetries with fixed singular values
Let $\left(\mathcal{M}^2,g_\mathcal{M};X\right)$ and $\left(\mathcal{N}^2,g_{\mathcal{N}};Y\right)$ be two smooth two-dimensional, simply connected Riemannian manifolds (with or without boundary), equ …
- 134
2
votes
1
answer
182
views
Signs of curvatures of integrals lines of frames with constant principal values
Let $D\subset\mathbb{R}^2$ be a planar domain (maybe simply connected) and consider all the mappings $f:D\to\mathbb{R}^2$ with constant, fixed, positive singular values. Let $E=(E_1,E_2)$ be the ortho …
- 134
2
votes
1
answer
117
views
Can we always find coordinates on a surface such that $K=K(u-v)$?
Let $(M^2,g)$ be a 2-dimensional Riemannian manifold. For any point $p\in M^2$ can we always find coordinates $(u,v)$ in a neighborhood $U$ of $p$ such that the Gaussian curvature is only a function o …
- 134
3
votes
1
answer
340
views
Shrinking a disk with fixed differential
Consider mappings $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ with differential
\begin{align}
\mathsf{d} f= \begin{pmatrix}
\cos\psi(x) &\cos\phi(y) \\
\sin \psi(x)& \sin\phi(y)
\end{pmatrix},
\e …
- 134
1
vote
Is there a volume-preserving diffeomorphism of the disk with prescribed singular values?
Did you find out the answer to the original question ?
I came across this* work (pg. 775, conjecture 7.1) where precisely that question is formulated as a conjecture (I ignore whether or not the autho …
- 134
3
votes
2
answers
221
views
$2\mathrm{d}$ area maximizing short embeddings
Think of a beach ball on an pool of water or sand.
Let $\left(\mathcal{M}^2,g\right)$ be a surface homeomorphic to a sphere, endowed with a Riemannian metric $g$, and $\left(\mathcal{N}^2,h\right)$ a …
- 134
5
votes
1
answer
365
views
Systems of (hyperbolic) 2nd order PDEs with lower order constraints
Certain surfaces in mechanics are endowed with the fundamental forms
\begin{align}
\text{I} &= \mathrm{d}u^2+\mathrm{d}v^2+2\cos\gamma\: \mathrm{d}u\: \mathrm{d}v \\
\text{II} &= \alpha\left(\gamma \r …
- 134