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Hilbert's fourth problem
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==Prehistory of Hilbert's fourth problem== [[Image:Cayley–Klein model.png|thumb|Cayley–Klein model of Lobachevsky geometry]] Before 1900, there was known the [[Cayley–Klein model]] of Lobachevsky geometry in the unit disk, according to which geodesic lines are chords of the disk and the distance between points is defined as a logarithm of the [[cross-ratio]] of a quadruple. For two-dimensional Riemannian metrics, [[Eugenio Beltrami]] (1835–1900) proved that flat metrics are the metrics of constant curvature.<ref>{{cite journal | last1=Beltrami | first1=Eugenio | authorlink1=Eugenio Beltrami | title=Risoluzione del Problema: Riportare i punti di una superficie sobra un piano in modo che le linee geodetiche Vengano rappresentate da linee rette | journal=Annali di Matematica Pura ed Applicata | volume=7 | date=1865 | pages=185–204| doi=10.1007/BF03198517 | s2cid=123192933 | url=https://zenodo.org/record/1595552 }}</ref> For multidimensional Riemannian metrics this statement was proved by [[Élie Cartan|E. Cartan]] in 1930. In 1890, for solving problems on the theory of numbers, [[Hermann Minkowski]] introduced a notion of the space that nowadays is called the finite-dimensional [[Banach space]].<ref>{{cite book | last1=Minkowski | first1=Hermann | authorlink1=Hermann Minkowski | title=Geometrie der Zahlen | publisher=B. G. Teubner, Leipzig-Berlin. | date=1953}}</ref> === Minkowski space === {{main|Minkowski space}} [[Image:Minkowski metric.svg|thumb|Minkowski space]] Let <math>F_{0}\subset \mathbb{E}^{n}</math>be a compact convex hypersurface in a Euclidean space defined by : <math>F_{0}=\{y\in E^{n}:F(y)=1\},</math> where the function <math>F=F(y)</math> satisfies the following conditions: # <math>F(y)\geqslant 0, \qquad F(y)=0 \Leftrightarrow y=0;</math> # <math>F(\lambda y)=\lambda F(y), \qquad \lambda\geqslant 0;</math> # <math>F(y)\in C^{k}(E^{n}\setminus \{0\}), \qquad k\geqslant 3;</math> # and the form <math> \frac{\partial^2 F^2}{\partial y^i \, \partial y^j}\xi^i\xi^j>0</math> is positively definite. The length of the vector ''OA'' is defined by: : <math>\|OA\|_M=\frac{\|OA\|_{\mathbb{E}}}{\|OL\|_{\mathbb{E}}}.</math> A space with this metric is called '''Minkowski space'''. The hypersurface <math>F_{0}</math> is convex and can be irregular. The defined metric is flat. ===Finsler spaces=== {{main|Finsler metric}} Let ''M'' and <math>TM=\{(x,y)|x\in M, y\in T_xM\}</math> be a smooth finite-dimensional manifold and its tangent bundle, respectively. The function <math>F(x,y)\colon TM \rightarrow [0, +\infty)</math> is called '''Finsler metric''' if # <math>F(x,y)\in C^{k}(TM\setminus \{0\}), \qquad k\geqslant 3</math>; # For any point <math>x\in M</math> the restriction of <math>F(x, y)</math> on <math>T_{x}M</math> is the Minkowski norm. <math>(M, F)</math> is '''Finsler space'''. === Hilbert's geometry === [[Image:Gilbert metric.svg|thumb|Hilbert's metric]] Let <math>U\subset (\mathbb{E}^{n+1}, \| \cdot \|_{\mathbb{E}})</math> be a bounded open convex set with the boundary of class ''C<sup>2</sup>'' and positive normal curvatures. Similarly to the Lobachevsky space, the hypersurface <math>\partial U</math> is called the absolute of Hilbert's geometry.<ref>{{cite journal | last1=Hilbert | first1=David | authorlink1=David Hilbert | title=Uber die gerade Linie als kürzeste Verbindung zweier Punkte | journal=Mathematische Annalen | volume=46 | date=1895 | pages=91–96 | doi=10.1007/BF02096204 | doi-access=free}}</ref> Hilbert's distance (see fig.) is defined by : <math>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}.</math> [[Image:Finsler metric.svg|thumb|Hilbert–Finsler metric]] The distance <math>d_{U}</math> induces the '''Hilbert–Finsler metric''' <math>F_{U}</math> on ''U''. For any <math>x\in U</math> and <math>y\in T_{x}U</math> (see fig.), we have : <math>F_U(x, y)=\frac{1}{2}\|y\|_{\mathbb{E}} \left( \frac{1}{\|x-x_{+}\|_{\mathbb{E}}}+\frac{1}{\|x-x_{-}\|_{\mathbb{E}}} \right). </math> The metric is symmetric and flat. In 1895, Hilbert introduced this metric as a generalization of the Lobachevsky geometry. If the hypersurface <math>\partial U </math> is an ellipsoid, then we have the Lobachevsky geometry. === Funk metric === In 1930, Funk introduced a non-symmetric metric. It is defined in a domain bounded by a closed convex hypersurface and is also flat.
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