Hilbert's fourth problem

Minkowski space

Let ${\displaystyle F_{0}\subset \mathbb {E} ^{n}}$be a compact convex hypersurface in a Euclidean space defined by

${\displaystyle F_{0}=\{y\in E^{n}:F(y)=1\},}$

where the function ${\displaystyle F=F(y)}$ satisfies the following conditions:

1. ${\displaystyle F(y)\geqslant 0,\qquad F(y)=0\Leftrightarrow y=0;}$
2. ${\displaystyle F(\lambda y)=\lambda F(y),\qquad \lambda \geqslant 0;}$
3. ${\displaystyle F(y)\in C^{k}(E^{n}\setminus \{0\}),\qquad k\geqslant 3;}$
4. and the form ${\displaystyle {\frac {\partial ^{2}F^{2}}{\partial y^{i}\,\partial y^{j}}}\xi ^{i}\xi ^{j}>0}$ is positively definite.

The length of the vector OA is defined by:

${\displaystyle \|OA\|_{M}={\frac {\|OA\|_{\mathbb {E} }}{\|OL\|_{\mathbb {E} }}}.}$

A space with this metric is called Minkowski space.

The hypersurface ${\displaystyle F_{0}}$ is convex and can be irregular. The defined metric is flat.

Let M and ${\displaystyle TM=\{(x,y)|x\in M,y\in T_{x}M\}}$ be a smooth finite-dimensional manifold and its tangent bundle, respectively. The function ${\displaystyle F(x,y)\colon TM\rightarrow [0,+\infty )}$ is called Finsler metric if

1. ${\displaystyle F(x,y)\in C^{k}(TM\setminus \{0\}),\qquad k\geqslant 3}$;
2. For any point ${\displaystyle x\in M}$ the restriction of ${\displaystyle F(x,y)}$ on ${\displaystyle T_{x}M}$ is the Minkowski norm.

${\displaystyle (M,F)}$ is Finsler space.

Hilbert's metric

Let ${\displaystyle U\subset (\mathbb {E} ^{n+1},\|\cdot \|_{\mathbb {E} })}$ be a bounded open convex set with the boundary of class C² and positive normal curvatures. Similarly to the Lobachevsky space, the hypersurface ${\displaystyle \partial U}$ is called the absolute of Hilbert's geometry.[9]

Hilbert's distance (see fig.) is defined by

${\displaystyle d_{U}(p,q)={\frac {1}{2}}\ln {\frac {\|q-q_{1}\|_{E}}{\|q-p_{1}\|_{E}}}\times {\frac {\|p-p_{1}\|_{E}}{\|p-q_{1}\|_{E}}}.}$

Hilbert–Finsler metric

The distance ${\displaystyle d_{U}}$ induces the Hilbert–Finsler metric ${\displaystyle F_{U}}$ on U. For any ${\displaystyle x\in U}$ and ${\displaystyle y\in T_{x}U}$ (see fig.), we have

${\displaystyle F_{U}(x,y)={\frac {1}{2}}\|y\|_{\mathbb {E} }\left({\frac {1}{\|x-x_{+}\|_{\mathbb {E} }}}+{\frac {1}{\|x-x_{-}\|_{\mathbb {E} }}}\right).}$

The metric is symmetric and flat. In 1895, Hilbert introduced this metric as a generalization of the Lobachevsky geometry. If the hypersurface ${\displaystyle \partial U}$ is an ellipsoid, then we have the Lobachevsky geometry.

In 1930, Funk introduced a non-symmetric metric. It is defined in a domain bounded by a closed convex hypersurface and is also flat.

Georg Hamel was first to contribute to the solution of Hilbert's fourth problem.[2] He proved the following statement.

Theorem. A regular Finsler metric ${\displaystyle F(x,y)=F(x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{n})}$ is flat if and only if it satisfies the conditions:

${\displaystyle {\frac {\partial ^{2}F^{2}}{\partial x^{i}\,\partial y^{j}}}={\frac {\partial ^{2}F^{2}}{\partial x^{j}\,\partial y^{i}}},\,i,j=1,\ldots ,n.}$

Consider a set of all oriented lines on a plane. Each line is defined by the parameters ${\displaystyle \rho }$ and ${\displaystyle \varphi ,}$ where ${\displaystyle \rho }$ is a distance from the origin to the line, and ${\displaystyle \varphi }$ is an angle between the line and the x-axis. Then the set of all oriented lines is homeomorphic to a circular cylinder of radius 1 with the area element ${\displaystyle dS=d\rho \,d\varphi }$. Let ${\displaystyle \gamma }$ be a rectifiable curve on a plane. Then the length of ${\displaystyle \gamma }$ is

$${\displaystyle L={\frac {1}{4}}\iint _{\Omega }n(\rho ,\varphi )\,dp\,d\varphi }$$

where ${\displaystyle \Omega }$ is a set of lines that intersect the curve ${\displaystyle \gamma }$, and ${\displaystyle n(p,\varphi )}$ is the number of intersections of the line with ${\displaystyle \gamma }$. Crofton proved this statement in 1870.[10]

A similar statement holds for a projective space.

In 1966, in his talk at the International Mathematical Congress in Moscow, Herbert Busemann introduced a new class of flat metrics. On a set of lines on the projective plane ${\displaystyle RP^{2}}$ he introduced a completely additive non-negative measure ${\displaystyle \sigma }$, which satisfies the following conditions:

1. ${\displaystyle \sigma (\tau P)=0}$, where ${\displaystyle \tau P}$ is a set of straight lines passing through a point P;
2. ${\displaystyle \sigma (\tau X)>0}$, where ${\displaystyle \tau X}$ is a set of straight lines passing through some set X that contains a straight line segment;
3. ${\displaystyle \sigma (RP^{n})}$ is finite.

If we consider a ${\displaystyle \sigma }$-metric in an arbitrary convex domain ${\displaystyle \Omega }$ of a projective space ${\displaystyle RP^{2}}$, then condition 3) should be replaced by the following: for any set H such that H is contained in ${\displaystyle \Omega }$ and the closure of H does not intersect the boundary of ${\displaystyle \Omega }$, the inequality

${\displaystyle \sigma (\pi H)<\infty }$ holds.[11]

Using this measure, the ${\displaystyle \sigma }$-metric on ${\displaystyle RP^{2}}$ is defined by

${\displaystyle |x,y|=\sigma \left(\tau [x,y]\right),}$

where ${\displaystyle \tau [x,y]}$ is the set of straight lines that intersect the segment ${\displaystyle [x,y]}$.

The triangle inequality for this metric follows from Pasch's theorem.

Theorem. ${\displaystyle \sigma }$-metric on ${\displaystyle RP^{2}}$ is flat, i.e., the geodesics are the straight lines of the projective space.

But Busemann was far from the idea that ${\displaystyle \sigma }$-metrics exhaust all flat metrics. He wrote, "The freedom in the choice of a metric with given geodesics is for non-Riemannian metrics so great that it may be doubted whether there really exists a convincing characterization of all Desarguesian spaces".[11]

The following theorem was proved by Pogorelov in 1973[3][4]

Theorem. Any two-dimensional continuous complete flat metric is a ${\displaystyle \sigma }$-metric.

Thus Hilbert's fourth problem for the two-dimensional case was completely solved.

A consequence of this is that you can glue boundary to boundary two copies of the same planar convex shape, with an angle twist between them, you will get a 3D object without crease lines, the two faces being developable.

Illustration of the Pogorelov Theorem

In 1976, Ambartsumian proposed another proof of Hilbert's fourth problem.[5]

His proof uses the fact that in the two-dimensional case the whole measure can be restored by its values on biangles, and thus be defined on triangles in the same way as the area of a triangle is defined on a sphere. Since the triangle inequality holds, it follows that this measure is positive on non-degenerate triangles and is determined on all Borel sets. However, this structure can not be generalized to higher dimensions because of Hilbert's third problem solved by Max Dehn.

In the two-dimensional case, polygons with the same volume are scissors-congruent. As was shown by Dehn this is not true for a higher dimension.