# Questions tagged [affine-geometry]

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70
questions

**37**

votes

**1**answer

2k views

### Is an affine fibration over an affine space necessarily trivial?

Let $X$ be an algebraic variety over an alg. closed field with zero char. and let $f:X\to \mathbb{A}^n$ be a smooth surjective morphism, such that all fibers (at closed points) are isomorphic to $\...

**17**

votes

**1**answer

513 views

### Covering the disk with a family of infinite total measure

Let $(U_n)_n$ be an arbitrary sequence of open subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum_{n=0}^\infty \lambda(U_n)=\infty$ (where $\lambda$ is the Lebesgue measure). Does ...

**15**

votes

**2**answers

946 views

### When is a flow geodesic and how to construct the connection from it

Let $(M,\Gamma)$ be a $C^\infty$ $n$ dimensional real manifold with a linear connection $\Gamma$ on it. I know the following:
If $\gamma:[t_0,t_1]\rightarrow M$ is a smooth curve and is a geodesic, ...

**14**

votes

**3**answers

817 views

### Can a convex polytope with $f$ facets have more than $f$ facets when projected into $\mathbb{R}^2$?

Let $P$ be a convex polytope in $\mathbb{R}^d$ with $n$ vertices and $f$ facets.
Let $\text{Proj}(P)$ denote the projection of $P$ into $\mathbb{R}^2$.
Can $\text{Proj}(P)$ have more than $f$ facets?
...

**14**

votes

**1**answer

469 views

### Projective-invariant differential operator

This question was originally asked on Math StackExchange.
Suppose we want a differential operator $T$ acting on functions $\mathbb{R}^n \rightarrow \mathbb{R}^n$ such that
\begin{align*}
&T(g) = ...

**12**

votes

**1**answer

332 views

### Covering the disk with a family of infinite total measure - the convex sequel

Let $(U_n)_n$ be an arbitrary sequence of open convex subsets of the unit disk $D(0,1)\subseteq \mathbb{R}^2$ s.t. $\sum_{n=0}^\infty \lambda(U_n)=\infty$ (where $\lambda$ is the Lebesgue measure). ...

**10**

votes

**3**answers

504 views

### 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 ...

**10**

votes

**2**answers

326 views

### Affine spaces as algebras for an operad?

(at.algebraic-topology because I don't know who else thinks about operads)
Let $A$ be an affine space, i.e. a torsor over (the abelian group underlying) a vector space $V$ over a field $K$. Then for ...

**10**

votes

**1**answer

1k views

### Who first proved the fundamental theorem of projective geometry?

The following theorem is often called the fundamental theorem of projective geometry:
Let $k$ be a field and let $n \geq 3$. Let $X$ be the partially ordered set of nonzero proper subspaces of $k^n$....

**10**

votes

**1**answer

475 views

### How many rich directions does a set in $\mathbb F_p^2$ determine?

$\newcommand{\F}{\mathbb F}$
A subset $P$ of the affine plane $\F_p^2$ is said to determine a direction if there is a line in this direction containing at least two points of $P$.
A set of size $|P|&...

**9**

votes

**2**answers

409 views

### Bieberbach theorem for compact, flat Riemannian orbifolds

In his thesis, Bieberbach solved Hilbert 18 problem and
proved that any compact, flat Riemannian manifold is a
quotient of a torus. I need a reference to an orbifold version
of this result: any ...

**9**

votes

**0**answers

235 views

### Almost blocking sets in $\mathbb F_q^2$

$\newcommand{\F}{{\mathbb F}}$
Let $q$ be an odd prime power. A blocking set in the affine plane $\F_q^2$ is a set blocking (meeting) every line.
A union of two non-parallel lines is a blocking set ...

**8**

votes

**1**answer

245 views

### Separating a lattice simplex from a lattice polytope

Let $P\subset\mathbb{R}^n$ be a convex lattice polytope.
Do there always exist a lattice simplex $\Delta\subset P$ and an affine hyperplane $H\subset\mathbb{R}^n$ separating $\Delta$ from the convex ...

**8**

votes

**1**answer

369 views

### Existence of certain "nondegenerate" function and manifold topology

Let $M$ be a smooth manifold without boundary, not necessarily compact.
Let $f$ be a real-valued smooth function on $M\times M$. We say $f$ is good if for any point $(x,y)\in M\times M$ with local ...

**7**

votes

**1**answer

249 views

### What are some compact Hessian manifolds?

In case this is too general, here is a more specific question.
Is there a hyperbolic threefold which admits a Hessian metric (hyperbolic or otherwise)?
Background
A Hessian manifold is a Riemannian ...

**7**

votes

**0**answers

280 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 ...

**6**

votes

**1**answer

239 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 ...

**5**

votes

**4**answers

2k views

### Projective to Affine?

Perhaps a basic question... how does one study affine algebraic geometry via projective geometry? For example, suppose I have two affine varieties which I want to prove are isomorphic, would it help ...

**5**

votes

**1**answer

205 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 ...

**5**

votes

**1**answer

371 views

### $(n-2)$-blocking sets in $AG(n,2)$

Let's define $k$-blocking set in affine space $AG(n,q)$ a set that meets every coset (translate of subspace) of dimension $k$.
I have seen a lot work related to minimal $(n-1)$-blockings set.
...

**5**

votes

**1**answer

182 views

### Affinely flat structures. How many different ones on the same manifold?

Let $M$ be a smooth manifold endowed with $\nabla$ a flat torsion-free connection. Let $\operatorname{Diff}(M)$ be the group of smooth diffeomorphisms of $M.$ Obviously if $ \phi \in \operatorname{...

**5**

votes

**2**answers

361 views

### Diameter-area ratio for affine tranformations.

Consider an convex plane figure $F$. How to prove that there is an affine transformation $a$ such that $\sqrt{3}$ diameter$(a(F))^2\leq 4$ area$(a(F))$?
I found only one reference, to "Über einige ...

**5**

votes

**1**answer

142 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 ...

**5**

votes

**1**answer

376 views

### What is meant by a Lie group acting by affine transformations?

This question is a continuation of this one, where I did not receive a complete answer so am moving to MO. I am trying to understand a paper by JP Szaro that was referred to there, and in particular, ...

**4**

votes

**2**answers

357 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{...

**4**

votes

**1**answer

255 views

### Square-free sets in $\mathbb F_2^n\oplus\mathbb F_2^n$

A square in $\mathbb F_2^n\oplus\mathbb F_2^n$ is a quadruple of the form
$$ (u,v)+\{(0,0),(0,d),(d,0),(d,d)\},\quad u,v,d\in\mathbb F_2^n,\ d\ne 0. $$
A set $A\subset\mathbb F_2^n\oplus\mathbb F_2^...

**4**

votes

**2**answers

134 views

### Affine hull of a set of non-negative matrices with fixed row-sums

Fix any non-negative matrix $M \in \mathbb{R}_{\geq 0}^{m \times n}$ that contains no zero-row and no zero-column.
Further, fix any positive vector $r \in \mathbb{R}_{> 0}^m$.
With $nz(M) := \{(i,j)...

**4**

votes

**1**answer

179 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 $C'$ ...

**4**

votes

**0**answers

266 views

### Osculating ellipsoids

Let $K$ be a given smooth, origin-symmetric, strictly convex body in $n$ dimensional euclidean space. At every point $x$ on the boundary of $K$ there exists an origin-symmetric ellipsoid $E_x$ that ...

**3**

votes

**3**answers

246 views

### Space of halfspaces

Consider a real vector space $V$ with dimension $n$ (say $V=\mathbb{R}^n$). The construction I'm describing is similar to the construction of the projective space. Instead of the space of lines, it is ...

**3**

votes

**1**answer

97 views

### Sets blocking every $2$-flat in $AG(n,2)$

The following may be well-known $-$ but not known to me:
What is the smallest possible size of a set in ${\mathbb F}_2^n$ that blocks every $2$-flat?
Here "blocks" means "have a non-empty ...

**3**

votes

**1**answer

60 views

### For which sets of $(n, m, k)$ does there exist an edge-labelling (using $k$ labels) on $K_n$, such that every single-labelled subgraph is $K_m$?

Or, equivalently - for what sets of $(n, m, k)$ is it possible, for a group* of $n$ people, to arrange $k$ days of "meetings", such that every day the group is split into subgroups of $m$ people, and ...

**3**

votes

**1**answer

270 views

### Cancellation problem for Laurent polynomial rings and power series rings

Throughout, let $k$ be an algebraically closed field. For two $k$-algebras $A,B$ let us write $A \cong_k B$ to mean that $A,B$ are isomorphic as $k$-algebras.
It is known that if $A$ is an integral ...

**3**

votes

**0**answers

132 views

### Barycenters and the axiomatic of affine geometry

There are several ways to define an affine space,
either by starting from a transitive action of a vector
space on a set of points, or listing sets of axioms
related to parallelism in the spirit of ...

**3**

votes

**0**answers

229 views

### Parallel Ricci condition - Status report and bibliography

First I'd like to point out that I'm not a mathematician but a physicist. Dealing with a (hopefully) new affine theory of gravity we have find that the equation of motion are not the usual Einstein's ...

**3**

votes

**0**answers

126 views

### Strictly Convex Smoothing of a function defined on an affine manifold

A function defined on an interval is strongly convex if it has positive definite second derivative. A function on an affine manifold is strongly convex
if its restriction to each line is. An affine ...

**3**

votes

**0**answers

687 views

### Theorems in affine geometry which can be proved using only dilations, generalized to metric spaces

Background: significant parts of E. Artin, Geometric algebra, Wiley-Interscience, New York, 1957 can be seen as consequences of the statement
(1) "the inverse semigroup generated by dilations in ...

**2**

votes

**1**answer

139 views

### Intersection of a $\mathbb{Q}$-affine space with $\mathbb{Z}^n$

Let $E$, a $\mathbb{Q}$-affine space of arbitrary dimension included in $\mathbb{Q}^n$. Is it possible to check efficiently if $E \cap \mathbb{Z}^n$ is empty or not?
If is an hard problem could give ...

**2**

votes

**1**answer

210 views

### If any two triangles of equal area can be mapped via affine maps, what can we say about the geometry?

This is a cross-post.
Let $(M,g)$ be a two-dimensional compact surface, endowed with a Riemannian metric.
Fix $s>0$, and suppose that for any two geodesic triangles $A,B$ having area $s$, there ...

**2**

votes

**1**answer

457 views

### Definition of Affine Root System: What is $\alpha_{+}$?

I was beginning to read Bruhat and Tits article Groupes Reductifs sur un Corps Locale and was confused on a point in the beginning of the section on affine root systems.
$\mathbf{A}$ is a finite ...

**2**

votes

**1**answer

94 views

### How does the affine Desargues theorem imply the little Desargues theorem?

Let $A$ be an (abstract) affine plane. We call $A$ a translation plane if the group of translations acts transitively on the set of points (Axiom 4a in Artin's book "Geometric algebra").
...

**2**

votes

**1**answer

283 views

### Reference for "topological affine spaces"

I am wondering if there is a topological version of affine spaces as a topological space along with a free transitive (continuous) action of a topological vector space on it?
Here is a notion so-...

**2**

votes

**0**answers

79 views

### Segre's theorem in $3$ dimensions with a "twist"

As I understand, there is a $3$-dimensional analogue of Segre's theorem stating that the maximum size of a set in ${\bf F}_q^3$ ($q$ odd) with no three points collinear is $q^2+1$. I am trying to ...

**2**

votes

**0**answers

93 views

### Lie correspondence for Ind groups

I have the following question (with a negative answer in general).
Consider an Ind algebraic group (e.g. a group object in category of ind varieties) as defined by Shafrevich. It was assumed for some ...

**2**

votes

**0**answers

243 views

### Flatness of modules over dual numbers

Let $X$ be a smooth, affine complex surface, and $M$ be a coherent $\mathcal{O}_X$-module. Denote by $D:=\mbox{Spec}(\mathbb{C}[t]/(t^2))$ and $X_D:=X \times_{\mathbb{C}} D$, the trivial deformation ...

**2**

votes

**1**answer

75 views

### Relation between the geodesics of Finsler norms $F(V)$ and $F(-V)$

I am trying to solve this exercise. Let $(M,F)$ be a Finsler space and define $\tilde{F}(x,y):=F(x,-y)$. Then $(M,\tilde{F})$ is a Finsler space and given a geodesic $t\mapsto \gamma(t)$ of $F$, $t\...

**1**

vote

**1**answer

366 views

### Pasch axiom and Pythagorean field condition?

I am looking for a reference for the claim that the Pasch axiom is equivalent to the Pythagorean field condition, and with respect to what base theory this should be true.
Since posting the question, ...

**1**

vote

**1**answer

122 views

### Polynomial isometries of $\mathbb{A}^2_\mathbb{C}$

I have the following question, which I'm sure must be explored somewhere.
Consider a group of polynomial automorphisms of $\mathbb{A}^2_\mathbb{C}$ preserving a standard hermitian metric. Is there any ...

**1**

vote

**1**answer

170 views

### Affine differential geometry. Is Calabi's hypersurface isotropic?

I am in the framework of (equi)affine differential geometry. Let $E$ be a centro-equiaffine space, that is a real vector space of dimension $n$, together with the special linear group $SL_n(R)$. Let $...

**1**

vote

**1**answer

230 views

### Automorphism group of fiber products of schemes

Let $A \mapsto S$ and $B \mapsto S$ be two schemes over the scheme $S$. Is there a connection between the automorphism group of the scheme $A \otimes_{S} B$ and the automorphism groups of $A$ and $B$ ?...