The study of geometry of manifolds without appealing to differential calculus. It includes studies of length spaces, Alexandrov spaces, and CAT(k) spaces. The techniques are often applicable to Riemannian/Finsler geometry (where differential calculus is used) and geometric group theory. For questions about plain-old metric spaces, please use (metric-spaces) instead.
Questions tagged [metric-geometry]
282 questions
1
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1 answer
Examples of CAT(0)-spaces that are no manifolds
I´m trying to think of some examples of CAT(0)-spaces that are not manifolds. I haven´t found any examples in the book of Bridson and Haefliger, so I have tried to come up with some own examples. I suppose there are some involving Graph Theory, but…
Saimel
- 138
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0 answers
$2$-dim positively curved Alexandrov space
A statement in the book I am reading states that any $2$-dimensional positively curved closed Alexandrov space is homeomorphic to $S^2$ or $\mathbb RP^2$. Is there any reference for this fact?
Totoro
- 829
1
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1 answer
Kuratowski convergence for unbounded/non-compact sets
Definition of Kuratowski convergence as copied from the Wikipedia page:
Let $(X,d)$ be a metric space, where $X$ is a set and $d$ is the function of distance between points in $X$.
For any $x \in X$ and any non-empty compact subset $A \subseteq X$,…
Marlonso
- 11
1
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0 answers
Angles made by a segment in two concentric circles
I have a quite tricky geometry problem to solve, and I don't find the answer. Here is the problem:
I have two concentric circles of known radii (r for the inner circle and R for the outer one). Between those circles, there is a segment which cuts…
Simon
1
vote
5 answers
Maximum number of points attaining acute angles in $\mathbb{R}^n$
In $\mathbb{R}^n$ consider three points
$v_i$. Here at $v_2$ the angle $\angle v_1v_2v_3$ is acute if it is
strictly smaller then $\frac{\pi}{2}$.
Note that in $\mathbb{R}^2$, one can find three points such that
all angles are acute.
For example,…
HK Lee
- 19,964
1
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1 answer
Geodesics in ultralimit of complete length spaces
Assume that $ \omega$ is a selective nonprinciple
ultrafilter. Fix $\omega$.
Def : Define $X_\omega$ to be set of equivalence
classes of sequence $(x_n),\ x_n\in X_n$ where $X_n$ is a metric
space. And we have a
metric $d_\omega$ on $X_\omega$
$$…
HK Lee
- 19,964
0
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0 answers
Why is the projection from the standard simplex to the sphere Lipschitz continuous?
In the book by Bridson and Häfliger, the following question is presented:
Prove that the identity map of any abstract simplicial complex induces a bi-Lipschitz homeomorphism between any two regular $M_\kappa$-simplicial complexes associated with $K$…
segi
- 1
- 1
0
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1 answer
Doubt about exercise on family of circles
Consider the following family of cirles
$$
x^2+y^2-2x+4y-2+k(x^2+y^2-6y+8)=0
$$
We are asked about the value of $k$ so that the circle will have the center on the straight line given by the equation
$$
y=-8x.
$$
The result given by the book is…
0
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0 answers
Intuition for finding a circumcenter using cross products?
The popular way to determine the circumcenter of a triangle is to find the intersection of two perpendicular bisectors, but I noticed the wikipedia page for circumscribed circle also describes a method involving cross products and dot products -- no…
prideout
- 111
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2 answers
Converting polar coordinates (degree) to Cartesian line coordinates
I am trying to calculate the (x1,y1), (x2,y2) coordinates of a line. From the image consider the rectangle of width w, height h, center c and angle θ.
If the same is given in a graph with X and Y axis then we can draw a tangent with angle θ from…
Sammy
0
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1 answer
Definition of uniformly bounded lengths of curves
On page 47 of ''A Course in Metric Geometry'' by Burago, Burago and Ivanov there is a version of Arzela-Ascoli theorem:
In a compact metric space, any sequence of curves with uniformly bounded lengths contains a uniformly converging subsequence.
In…
