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Results tagged with alexandrov-geometry
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user 1441
Alexandrov geometry studies non smooth analogues of Riemannian manifolds with curvature bounded from below or above. It includes spaces with curvature bounded below (briefly $\mathrm{CBB}[\kappa]$) and spaces with curvature bounded above (briefly $\mathrm{CAT}[\kappa]$).
4
votes
Accepted
The set of strained points in an Alexandrov space is open
Note that this is an open condition, which depends on the distances $|a_ia_j|$,
$|a_ib_j|$, $|b_ib_j|$, $|pa_i|$ and $|pb_i|$, hence the result.
- 45k
9
votes
Accepted
Can Alexandrov surfaces of CAT(0) type be approximated by CAT(0) polyhedra?
I am sure it done somewhere, but I do not know a ref. I did something like this in my "Metric minimizing surfaces", but do not want to claim originality.
You may fix a finite set of points draw all t …
- 45k
7
votes
3-dim positively curved Alexandrov space
I guess you are interested in topological classification (?).
Given a 3-dimensional Alexandrov space $M$,
you can always find an other Alexandrov space $\bar M$ with isometric involution $J$
such tha …
- 45k
2
votes
Toponogov's theorem in the Alexandrov space with respect to a compact set
Yes it is true. It can be proved by applying distance estimates to the gradient flow of the function $f=\mathrm{dist}_A^2/2$. See our draft.
- 45k
5
votes
Accepted
Is any geodesic in the tangent cone of an Alexandrov space a limit geodesic?
The answer is "yes" assuming you are interested in minimizing geodesics.
Moreover the statement holds in the collapsing case as well.
Fix two points $p$ and $q$ in the limit space $X$.
If there is un …
- 45k
3
votes
Accepted
Two geodesics with angle $\pi$ in Alexandrov space
Yes, it is possible.
Consider the graph $\Gamma$ of
$$z=\phi\left(\sqrt{x^2+y^2}\right),$$ where $\phi\colon\mathbb R\to \mathbb R$ is a convex even function with $\phi(0)=0$.
With its intrinsic metr …
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1
vote
Coordinate chart of concave functions near a regular point in Alexandrov spaces
The needed construction was given by Perelman and it is described in many places, for example 7.11 in my "Semiconcave functions...".
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8
votes
Accepted
Whether the manifold part of an Alexandrov space is connected?
Yes.
Assume $A$ is an $m$-dimensional Alexandrov space and $\Omega\subset A$ be the maximal open subset which is a topological $m$-manifold and $A'\subset A$ be the subset of all points with tangent …
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5
votes
Accepted
positively curved Alexandrov space
Again, I guess you want a topological classification.
Such classification would include classification of all smooth positively curved manifolds
which is too much to ask.
For the (quasi)geodesic subs …
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1
vote
about parabolic cone
The metric of warp products $F\times_f B$ is fiber independent.
It means that if $(x,a),(y,b)\in F\times_f B$ and $(x',a),(y',b)\in F'\times_f B$
then
$$|x-y|_F=|x'-y'|_F
\ \ \ \Rightarrow\ \ \
|(x …
- 45k
3
votes
examples of space of direction at a point in an infinite dim Alexandrov space compact
Yes, the space of direction of an infinite dim Alexandrov space can be compact at some point.
Take for example the pyramid with Hilbert cube as the base.
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3
votes
Soul theorem for non-negativly curved open Alexandrov manifolds?
It might be true. If I would have to prove it I would try to mimic standard proof in Riemannian geometry --- Take $f$ to be distant from the soul. If $f$ has no critical points then you space is homeo …
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5
votes
Accepted
Are isometries the only geodesic preserving maps in a CAT(0)-space?
The map which you call "geodesic preserving" is usually called "affine".
It seems that affine maps to the real line are well understood even for general length space.
For your later edit: you may alw …
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3
votes
Accepted
Is Level set of Regular functions in Alexandrov spaces again an Alex. space?
The answer is "no" even for regular semiconcave function $f:X\to\mathbb R$
If $f:X\to \mathbb R$ is convex then it is a long-standing open problem.
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7
votes
Accepted
Is there Domain Invariance for Alexandrov spaces?
The following lemma from Grove--Petersen, A radius sphere theorem does the trick.
Lemma 1. Let $X$ be a compact Alexandrov space without boundary. Then $X$ has a fundamental class in Alexander-Spa …
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