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 be $M$ ${\ displaystyle M}$ - connected (possibly not complete ) Riemannian manifold of dimension $\geq 3$ ${\ displaystyle \ geq 3}$ . If the sectional curvature $K_{\sigma _{p}}$ ${\ displaystyle K _ {\ sigma _ {p}}}$ where $\sigma _{p}$ ${\ displaystyle \ sigma _ {p}}$ there is a plane in $T_{p}(M)$ ${\ displaystyle T_ {p} (M)}$ depends only on $p$ ${\ displaystyle p}$ then $M$ ${\ displaystyle M}$ 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.

Schur theorem on constant curvature

Formulation

Literature

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