Фінслерова геометрія

Фінслерова геометрія — одне з узагальнень ріманової геометрії. У фінслеровій геометрії розглядаються многовиди з фінслеровим метричним тензором; тобто вибором гладкої норми на кожному дотичному просторі, яка гладко змінюється від точки до точки.

Основні поняття

Нехай $M^{n}$ — $n$ -вімірний зв'язаний $C^{\infty }$ -многовид. Позначимо через $TM^{n}$ дотичне розшарування $M^{n}$ . Тоді фінслеровою метрикою на $M^{n}$ називається функція $F\colon TM^{n}\rightarrow [0,\infty )$ , що задовільняє властивостям:

$F\in C^{\infty }(TM^{n}\setminus \{0\})$ ;
$F$ додатньо однородна першої степени, тобто для будь-якої пари $(x,y)\in TM^{n}$ і числа $\lambda >0$ ,
$\ F(x,\lambda y)=\lambda F(x,y)$ ;
Для будь-якої пари $(x,y)\in TM^{n}$ білінійна форма $\mathbf {g} _{y}\colon T_{x}M^{n}\times T_{x}M^{n}\rightarrow \mathbb {R}$ ,
$\mathbf {g} _{y}(u,v)={\frac {1}{2}}{\frac {\partial ^{2}}{\partial t\,\partial s}}\lbrack F^{2}(x,y+su+tv)\rbrack |_{s=t=0}$

додатньо визначена.

Джерела

H. Rund. The Differential Geometry of Finsler Spaces, Springer-Verlag, 1959. ASIN B0006AWABG.

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