# Questions tagged [affine-geometry]

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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 ...
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{... 1answer 299 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, ... 1answer 284 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, ... 0answers 80 views ### 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 ... 1answer 171 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{... 2answers 116 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)... 0answers 115 views ### 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 ... 1answer 298 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. ... 0answers 115 views ### 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... 0answers 227 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 ... 0answers 139 views ### 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 ... 1answer 159 views ### 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 ... 1answer 960 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.... 0answers 217 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 ... 1answer 359 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 ... 1answer 158 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 ... 0answers 121 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 ... 0answers 148 views ### Compactification of the affine space with a del Pezzo surface Is there some nice compactification of the affine space with a del Pezzo surface of degree \le 4 (for example with a cubic surface) ? More precisely, I would like a projective algebraic variety X ... 1answer 156 views ### Is there a way to make MAGMA work with surfaces over weighted projective spaces? Is there a way to use MAGMA to study surfaces defined over a weighted projective space (by "study" I mean computing e.g. invariants (e.g. p_a, p_g), singularities, etc)? For example, I was trying, ... 0answers 620 views ### Does the coordinate ring of affine variety admit a structure of infinite dimensional variety? We work in the category of algebraic varieties over some algebraically closed field k. By infinite dimensional variety I mean a filtration:$$ V_0\subset V_1\subset V_2\subset\ldots  where each $... 3answers 464 views ### Line-preserving bijection of${\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$be an affinity? Assuming continuity would ... 1answer 1k views ### Is an affine fibration over an affine space necessarily trivial? Let$X$be an algebraic variety over an alg. closed field with zero char. and let$f:X\to \mathbb{A}^n$be a smooth surjective morphism, such that all fibers (at closed points) are isomorphic to$\...
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 ...