RESEARCH PROBLEM On generalizations of Hilbert’s fourth problem for

pseudo-Deasargues metrics

Tibor Ódor odormath-inst.hu http://www.math-inst.hu/˜ odor

November 16, 2000

Abstract We generalize Hilbert’s fourth problem for pseudo-Desarguas geome-

tries with symmetric metrics.

1 Introduction

As usual, R denote the field of real numbers, Z the ring of integers, R^{n} the n dimensional real vector space, O the origin in R^{n}, S^{n−1} the unit sphere with centre O and P^{n} the n dimensional projective geometry, respectively.

A convex body is a convex compact set in P^{n} with non void interior.

Let K ⊂ P^{n} be an open convex set in the n dimensional projective space P^{n}. We say that C_{0} ⊂ R^{n} is a non degenerated convex cone, if

(1) it is a closed set in R^{n} and it has non void interior,

(2) if the associated cone C = C_{0} ∪ −C_{0} does not contain two planes,

(3) if x, y ∈ C_{0} then x + y ∈ C_{0} too.

Let C_{0} be a continuous convex cone bundle in T^{∗}K and C is the associated bundle.

For every different points x, y ∈ K let t_{x} ∈ T_{x}K and t_{y} ∈ T_{y}K be the tangent vectors of the segment xy in the points x and y, respectively.

We say that the associated cone bundle C is relativistic, if for every different points x, y ∈ K holds that t_{x} ∈ C_{x} implies t_{y} ∈ C_{y}.

(It is clear that for every different x, y ∈ K if the continuous associated cone bundle is relativistic then t_{x} ∈ ∂C_{x} or t_{x} ∉ C_{x} implies t_{y} ∈ ∂C_{y} or t_{y} ∉ C_{y} too, respectively.)

If for the different points x, y ∈ K holds that t_{x} ∈ int C_{x}, t_{x} ∉ C_{x} or t_{x} ∈ ∂C_{x} we say that the segment xy is timelike, spacelike or lightlike, respectively.

Lemma 1 If K = R^{n} then for every relativistic associated cone bundle C holds that C_{x} − x = C_{y} − y.

Proof. The statement is essentially trivial.

Definition. A continuous function d : K × K → R is called a pseudo-Desargues metric admitting the relativistic associated cone bundle C if the following assumption holds:

(α) For every x, y ∈ K holds that d(x, y) = d(y, x),

(β) For every timelike, spacelike or lightlike segment xy ⊂ K holds that d(x, y) > 0, d(x, y) < 0 or d(x, y) = 0, respectively.

(γ) For every timelike or spacelike segment xy the shortest timelike or spacelike curve connecting x and y is the segment xy, respectively.

The main problem of the subject is as follows.

Problem 1 Describe all pseudo-Desargues metric on K.

We say that the convex cone C_{0} is timelike, if there is a closed convex set T ⊂ int C_{0} and a closed set S ⊂ out C with the following properties:

(A) For every line l ⊂ C not running in the boundary of C l ∩ T is a single point.

(B) For every hyperplane H not intersecting int C the closure of the set H ∩ out S is a convex body in H.

Conjecture 1 For every n ≥ 2 a timelike convex cone C_{0} is ellipsoidal.

Conjecture 2 If a relativistic associated cone bundle C admits a pseudo-Desargues metric then it is timelike.

Theorem 1 If a relativistic associated cone bundle C admits a C^{1} pseudo-Desargues metric, then it is timelike.

Proof. The proof is essentially trivial.

2 Acknowlwdgements

I am indebted to my teacher Zoltán Imre Szabó for introducing me into the subject and for the valuable discussion and information.

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