Questions tagged [affine-geometry]

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Geometry in $\mathbb{R}^n$: angle between projections of a rectangle

Consider a hyper rectangle $R$ in $\mathbb{R}^n$ defined by $|x_i|\leq M_i$ for all $i\leq n$. Consider a linear affine subspace $L$ of dimension $1\leq k <n$ such that $L\cap R\neq \emptyset$. For ...
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Gateaux differentiability of the norm in Banach spaces

I'm struggling to understand a particular implication in the proof of Corollary 5 of this paper involving Gateaux differentiability of the norm. The claim is that Gateaux differentiability of the norm ...
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Potentially elementary question on affine functions on Banach spaces

In Measures Which Agree On Balls by Hoffmann-Jørgensen, it is claimed that the function defined on $T(x)$, the set of normals to the unit sphere at $x$, given by $ \varphi(x^*) = \left\{ \begin{array}{...
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Comparative between Kobayashi hyperbolicity and hyperbolicity in the sense of Koszul

In literature, there are several notions of hyperbolicity. My question is whether, for closed locally flat or affine manifolds, the notion of hyperbolicity in the sense of Kobayashi is equivalent to ...
Samir's user avatar
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Another implication of the Affine Desargues Axiom

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
Taras Banakh's user avatar
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A corollary of the affine Desargues axiom

Definition 1. An affine plane is a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of subsets of $X$ called lines which satisfy the following axioms: Any distinct points $x,y\...
Taras Banakh's user avatar
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Points on affine hypersurface over finite field

I am interested in the hypersurface $X\subset\mathbb{A}^4_{\mathbb{F}_{5^n}}$ defined by $$ X = \{x^3 + 3xy^2 + z^3 + 3zw^2 + 1 = 0\} $$ over a finite field $\mathbb{F}_{5^n}$ with $5^n$ elements. Via ...
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3 answers
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Was the small Desargues Theorem known to ancient Greeks?

My question concerns the classical Desargues Theorem and its simplest version The small Desargues Theorem: Let $A$, $B$, $C$ be three distinct parallel lines and $a,a'\in A$, $b,b'\in B$, $c,c'\in C$,...
Taras Banakh's user avatar
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What does it mean "parallel"?

I am thinking on a strict definition of the notion of parallel affine sets in a linear space and came to the following Definition 1: An affine set $A$ is parallel to an affine set $B$ in a linear ...
Taras Banakh's user avatar
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Does Playfair imply Proclus?

I am interested in the interplay between the Playfair and Proclus Axioms in linear spaces. By a linear space I understand a pair $(X,\mathcal L)$ consisting of a set $X$ and a family $\mathcal L$ of ...
Taras Banakh's user avatar
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On convex solids with all plane sections affine congruent

Question: How many (classes of) convex 3D solids are there such that all non-degenerate planar sections of the solid are mutually affine congruent? Further question: Same as above with 'projective' ...
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Hypercube and affine space [closed]

We have an affine subspace $A$ of dimension $d_{A}$ and a hypercube $C$ of dimension $d_{C}$ ($d_{C}=n$), with $d_{A}\lt d_{C}$ and both belong to $\Re ^{n}$. Each face of dimension $d_{C}-1$ of $C$ ...
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Cutting convex polygons into affine-equivalent quadrilaterals

We continue from Tiling the plane with quadrilaterals that are mutually non-congruent and affine equivalent Question: Can any convex polygon $C$ be partitioned into some finite number $m$ of ...
Nandakumar R's user avatar
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Tiling the plane with quadrilaterals that are mutually non-congruent and affine equivalent

Question: Can the plane be tiled with convex quadrilaterals that are (1) mutually non-congruent in a Euclidean sense and (2) mutually affine-equivalent? Remark: Every trapezoid is affine equivalent to ...
Nandakumar R's user avatar
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1 answer
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Set of points covered by subspaces of small dimensions

Let $S \subset \mathbb R^d$ be finite set of points. We say that $S$ is $2$-covered if $S$ lies in a union $V_1\cup V_2$ of affine subspaces such that $\dim(V_1)+\dim (V_2)\leq d-1$. For example, if $...
Hailong Dao's user avatar
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1 answer
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How many ways are there to pick $n$ points on the finite affine plane $(\Bbb F_q)^2$ such that no three are collinear?

How many ways are there to pick $n$ points on the finite affine plane $(\Bbb F_q)^2$ such that no three are collinear? For example, how many ways can we pick $5$ points on $\Bbb F_{32}\times\Bbb F_{32}...
Akiva Weinberger's user avatar
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1 answer
219 views

Is the affine geometry a geometry of proportions?

Given any linear space $L$ over an ordered field $F$, consider the equiproportion relation $${\sim}=\{((x,y,z),(a,b,c))\in L^3\times L^3: \exists t\in[0,1]\subseteq F\;(y{-}x=t(z{-}x)\wedge b{-}a=t(c{-...
Taras Banakh's user avatar
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Intersection of the general linear group $\text{GL}_n(\mathbb{F}_q)$ and an affine subspace over a finite field

Let $\mathbb{F}_q$ be a finite field of $q$ elements, let $\text{M}_n(\mathbb{F}_q)$ be the vector space of $n \times n$ matrices over $\mathbb{F}_q$, let $\text{GL}_n(\mathbb{F}_q)$ be the group of $...
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3 votes
0 answers
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Projective plane finite game

This is a 2-person game. Let $\ P\ $ be any arbitrary projective space (of any dimension $\ \ge2$ and any cardinality, etc., but typically, let it be a finite plane over a field). Let $\ S_0\subseteq ...
Wlod AA's user avatar
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Can quasi affine varieties contain projective curves [closed]

Can a regular quasi affine variety (i.e. open subscheme of an affine variety) contain a (possibly singular) projective curve?
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Manifolds whose tangent spaces have a special behavior

Consider an $n$-dimensional complex manifold $M\subset\mathbb{C}^N$ and let $$f:\mathcal{U}\subset\mathbb{C}^n\rightarrow \mathcal{V}\subset M\subset\mathbb{C}^N$$ be a local parametrization of $M$. ...
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Anti-flag transitive affine planes

Let $\mathcal{A}$ be an axiomatic affine plane. First let $\mathcal{A}$ be finite. Suppose that the automorphism group of $\mathcal{A}$ acts transitively on nonincident point-line pairs (that is, on ...
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Maximal number of vertices of the intersection of a flat and a hypercube

Consider the intersection of an $n$-dimensional hybercube and an $m$-dimensional flat (affine subspace) which contains the diagonal of the hypercube. This is a convex polytope. What is the maximal ...
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Existence of an open set with specific properties

I am currently reading Gerard Laumon's Faisceaux caractères, which details various constructions and properties of character sheaves on a connected reductive group $G$ over $\overline{\mathbb{F}}_p$. ...
Martin Skilleter's user avatar
5 votes
1 answer
226 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"). ...
Brauer Suzuki's user avatar
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1 answer
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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 ...
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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 ...
Marcel K. Goh's user avatar
7 votes
1 answer
677 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 ...
Gabe K's user avatar
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1 vote
0 answers
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How to prove continuity in topological group action of ${\rm{GA}}_a(X)$ on $T(X)$, to make ${\rm{GA}}(X)$ a topological group?

The question comes from the following paragraph of a text on geometry in the context of affine geometry (Marcel Berger et al., "Geometry I", P56-57): 2.7.1.3. If we don't want to resort to ...
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1 answer
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Ellipsoid in $L^p([0,1],\lambda)$ spaces?

Let us consider $L^p([0,1],\lambda)$ spaces, were $\lambda$ is simply the lebesgue measure. These are Banach spaces for $p\ge1$ (of course). It is well known that for $ 1\leq p < q \leq +\infty$ we ...
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How to check if there exists a non-negative element in an affine space?

Suppose that a $n$-dimensional affine space is defined with vectors $e_i=(0,\dots,1,\dots,0), i=1,\dots,n$ (the $1$ stands in the $i$-th place), i.e. $$\mathcal{A}=\left\{\alpha_1e_1+\alpha_2e_2+\dots+...
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1 answer
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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 ...
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2 votes
0 answers
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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 ...
Grisha Taroyan's user avatar
1 vote
1 answer
134 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 ...
Grisha Taroyan's user avatar
3 votes
1 answer
71 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 ...
scubbo's user avatar
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1 vote
1 answer
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On a condition on ideals viwed as a Zariski open condition on co-tangent space

Let $(R, \mathfrak m,k)$ be a Noetherian local ring such that the residue field $k$ is infinite. Let $n=\mu(\mathfrak m)$. Then $n=\dim_k(\mathfrak m/\mathfrak m^2)$ . By fixing $x_1,...,x_n \in \...
uno's user avatar
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0 answers
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One parameter change of a section of $T^*M \otimes End(TM)$ on an affinely flat manifold

Let $\nabla$ be a flat symmetric connection in the tangent bundle of a smooth manifold $M.$ Let $A$ be a global section of $T^*M \otimes End(TM).$ Let $\phi:B(0,1) \to M$ be a local affine chart on $...
Mike Cocos's user avatar
2 votes
1 answer
173 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 ...
Ievgeni's user avatar
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13 votes
1 answer
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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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17 votes
1 answer
562 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 ...
5th decile's user avatar
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3 votes
0 answers
176 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 ...
coudy's user avatar
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5 votes
1 answer
172 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 ...
asv's user avatar
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1 vote
0 answers
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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 ...
Espace' etale's user avatar
2 votes
1 answer
403 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-...
ARA's user avatar
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9 votes
2 answers
615 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 ...
Misha Verbitsky's user avatar
2 votes
0 answers
384 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 ...
Chen's user avatar
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14 votes
1 answer
546 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) = ...
user76284's user avatar
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6 votes
1 answer
365 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 ...
Dox's user avatar
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1 vote
0 answers
177 views

Bijective correspondence between $\mathbb G_a$ actions on affine varieties and exponential maps on affine $k$-domains

Let $A$ be an integral domain which is a finitely generated algebra over an algebraically closed field $k$. Let $\phi :A \to A^{[1]}$ be a $k$-algebra homomorphism and let us write $\phi_t : A \to A[...
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1 vote
0 answers
150 views

Factorially closed, finitely generated $k$-sub-algebra of $k[X_1,X_2,X_3]$ , where $k$ is algebraically closed field of positive characteristic

Let $S$ be a sub-ring of a commutative ring with unity $R$. Then $S$ is called factorially closed in $R$ if $a,b \in R$ and $ab \in S\setminus \{0\} \implies a,b \in S$. Let $k$ be an algebraically ...
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