Теорема порівняння Берже — Каждана

Теорема порівняння Берже — Каждана — результат в римановій геометрії. Теорема дає точну нижню оцінку на об'єм ріманового многовиду, в термінах радіусу ін'єктивності, при цьому в разі рівності многовид ізометрічний стандартній сфері.

Теорема названа на честь Марселя Берже і Джеррі Каждана.

Формулювання

Нехай (M,g) — компактний m-вимірний ріманів многовид з радіусом ін'єктивності хоча б . Тоді об'єм (M,g) не менше об'єму m-вимірної одиничної сфери , а у випадку рівності (M,g) ізометричний .

Література

  • Berger, Marcel; Kazdan, Jerry L. (1980). A Sturm–Liouville inequality with applications to an isoperimetric inequality for volume in terms of injectivity radius, and to Wiedersehen manifolds. Proceedings of Second International Conference on General Inequalities, 1978. Birkhauser. с. 367—377. 
  • Kodani, Shigeru (1988). An Estimate on the Volume of Metric Balls. Kodai Mathematical Journal. 11 (2): 300—305. doi:10.2996/kmj/1138038881. Архів оригіналу за 8 січня 2019. Процитовано 2 квітня 2022. 

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