All Questions
7 questions
9
votes
2
answers
499
views
There is no arcwise isometry from a high dimensional manifold into a low dimensional manifold
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Let $X,Y$ be Riemannian manifolds, such that $\dim(X) > \dim(Y)$.
I am trying to prove the following statement (...
- 6,741
8
votes
1
answer
320
views
Does nonexpanding map between manifolds decrease volume?
(This question is a special case of a question I asked at SE, which got no answer there)
Let $M,N$ be diffeomorphic connected compact Riemannian manifolds, and let $f:M \to N$ be a surjective ...
- 6,741
7
votes
1
answer
373
views
Are metric isometries smooth at the boundary?
Let $M,N$ be smooth Riemannian manifolds with boundary (In particular, we assume the boundaries are smooth).
Suppose we have a map $\phi:M \to N$ which satisfies the following properties:
$$(1) \, \,...
- 6,741
7
votes
0
answers
669
views
Homometric $\Rightarrow$ isometric?
Suppose you know that there is a mapping between
two Riemmanian manifolds $M_1$ and $M_2$ such that,
for each $x_1 \in M_1$, the (codimension-1) measure of the set of points
at distance $d$ from $x_1$ ...
- 151k
6
votes
1
answer
185
views
Cohn-Vossen rigidity theorem in hyperbolic space
There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf
Any isometry between two closed smooth convex surfaces in ...
- 21.8k
3
votes
1
answer
454
views
Pogorelov's rigidity theorem vs Cohn-Vossen rigidity theorem
There is the following rigidity theorem of Cohn-Vossen as stated on p. 86 of these lecture notes: http://www.math.brown.edu/~deigen/chern.pdf
Any isometry between two closed smooth convex surfaces (...
- 21.8k
1
vote
0
answers
187
views
Does there exist an isometry between a regular polygon and a circle?
In order to define the question in a meaningful fashion, I am referring to a smooth manifold $\mathcal{M}$ within an $\epsilon$-neighborhood of a regular polygon $\mathcal{P}$ satisfying $$\max\{\|x-p\...
- 547