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Schur theorem on constant curvature

Schur's theorem gives a pointwise condition on the Riemannian metric, which guarantees the constancy of its curvature. Proved by Friedrich Schur in 1886.

Formulation

Let beM {\ displaystyle M}   - connected (possibly not complete ) Riemannian manifold of dimension≥3 {\ displaystyle \ geq 3}   . If the sectional curvatureKσp {\ displaystyle K _ {\ sigma _ {p}}}   whereσp {\ displaystyle \ sigma _ {p}}   there is a plane inTp(M) {\ displaystyle T_ {p} (M)}   depends only onp {\ displaystyle p}   thenM {\ displaystyle M}   there is a space of constant curvature.

Literature

  • with. 192, Sh. Kobayashi, K. Nomizu, Basics of Differential Geometry (inaccessible link)
  • Schur F. Über den Zusammenhang der Räume konstanter Krümmungsmasses mit den projektiven Räuraen , Mathematische Annalen , 1886. 27, S. 537-567.
Source - https://ru.wikipedia.org/w/index.php?title=Teorema_Shura_o_permanent_course&oldid=100521166


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