Questions tagged [affine-geometry]

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37
votes
1answer
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
1answer
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
2answers
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
3answers
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
1answer
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
1answer
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
3answers
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
2answers
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
1answer
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
1answer
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
2answers
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
0answers
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
1answer
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
1answer
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
1answer
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
0answers
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
1answer
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
4answers
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
1answer
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
1answer
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
1answer
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
2answers
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
1answer
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
1answer
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
2answers
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
1answer
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
2answers
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
1answer
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
0answers
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
3answers
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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
0answers
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
0answers
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
0answers
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
1answer
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
1answer
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
1answer
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
1answer
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
1answer
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 $B$ ?...