# Questions tagged [affine-geometry]

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### Probability that random points are affine subspace

I asked this question in Math stack exchange, but I think it is more relevant here. Suppose that $\mathbb{F}_q$ is a finite field with $q$ elements. Let $U = \{u_1, \ldots,u_m\}$ be a set of $m$ ...
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### 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 ...
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### 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). ...
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### 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 ...
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### 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 ...
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### 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 ...
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### complexity of system of equations defining affine variety

Say you have an affine variety $X$ in $n$-dimensional affine space. (You can even assume we are over $\mathbb{C}$, but I believe the nature of my question is algebraic). I want to bound from above ...
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### 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-...
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### 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 ...
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### 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 ...
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### 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) = ...
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### 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 ...
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### If a partial normalisation of an affine variety is affine?

Let $X$ be an affine variety. Let $x$ be an isolated singular point of $X$. Let $U$ is an affine neighbourhood of $x$ such that $U\setminus \{x\}$ is smooth. Let $\pi:\tilde{U}\to U$ be the ...
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### 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 ...
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### 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 ...
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### 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 $B$ ?...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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{...
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### 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, ...
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### 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, ...
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### Characterizing normal vectors of affinely independent subsets of the hypercube

Suppose we are looking at the hypercube in $\mathbb{R}^n$. I tend to the think of the cube with vertex coordinates 0 or 1, but maybe this is easier for the $\pm1$ cube. Now suppose we have an affine ...
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### What is this 2-form on a Lagrangian torus fibration?

Suppose we are given a regular $2n$-dimensional Lagrangian fibration $\pi : (M,\omega) \to B$ with connected, compact fibers. Then it is well-known (Arnold-Liouville) that each fibre is a Lagrangian ...
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### $(n-2)$-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. ...
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### Affine-regular hexagon in convex body

An affine-regular $n$-gon is a non-degenerate affine image of the regular $n$-gon. It seems to be a standard fact in combinatorial geometry that inside every convex compact set $K\subseteq \mathbb R^2$...
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### 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 ...
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### A questions related to the Markus conjecture for special affine manifolds

An affine manifold $M$ is called special if there is a parallel volume form $\omega$ on $M$, and a nowhere vanishing vector field $\mathcal{V}.$ Here we need to point out that any affine manifold of ...
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### Uniqueness affine curvature

Let $\gamma_1,\gamma_2: \mathbb{S}^1 \to \mathbb{R}^2$ be two smooth, closed, convex curves that their (special)affine curvature, $\mu_1,\mu_2$ are equal, that is $\mu_1(\theta)=\mu_2(\theta)$, for ...
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### 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 $k^n$....
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### 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 ...
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### 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 ...
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 \$...