# All Questions

9 questions
Filter by
Sorted by
Tagged with
82 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 ...
125 views

### Is a symmetric, parallel (0,2)-tensor a metric?

I'm interested in affinely connected spaces, on which a metric is not necessarily defined, i.e. $(\mathcal{M},\Gamma)$. Since (as a physicist) my goal is to consider a generalized model of gravity, I ...
157 views

### What conditions are sufficient to prove a transformation in $\mathbb{R}^2$ is affine? [closed]

What conditions are sufficient to prove a transformation in $\mathbb{R}^2$ is affine? I currently have shown that the transformation is bijective, points and lines are preserved and also the ...
174 views

### Maximin diameter of a transformed convex figure

You are given a compact convex figure in the plane, $C$. Your goal is to transform $C$ to a different figure $C'$ with the smallest possible diameter, as long as: The trasformation from $C$ to $C'$ ...
235 views

### Problem on triangles

Let $T\subset \mathbb{R}^2$ be any triangle and $T^t$ a deformation of $T$. Call $l_1,l_2,l_3$ the squares of the lengths of the sides of $T$ and $l_1^t,l_2^t,l_3^t$ the squares of the lengths of the ...
258 views

### Status of an open question in Artin's “Geometric Algebra”

In Artin's book "Geometric Algebra", Chapter II, the author states some axioms for geometry (section 1) and then begins to prove some results about the symmetries of the geometry (section 2). The ...
257 views

### How to show the two convex bodies are affinely isomorphic?

This problem comes from the response of the author of papers. Consider two convex bodies $A$ and $B$: $$A= \{X\in \mathcal{S}^4 : \operatorname{tr}(X) = 1, X\succeq 0 \}$$ B = \operatorname{...
### Line-preserving bijection of ${\mathbb{R}}^n$ onto itself
If $f:{\mathbb{R}}^n\to{\mathbb{R}}^n$ $(n\ge2)$ is a bijection such that the image of every line is a line (continuity of $f$ not assumed), must $f$ be an affinity? Assuming continuity would ...