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I am looking for concrete examples of negatively curved constant curvature manifold. The only example of negatively curved constant curvature manifold is the hyperbolic plane. Are there any easy examples of such manifolds which are compact.

zach
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1 Answers1

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Any compact orientable surface of genus $>1$ carries a metric with constant negative curvature

HK Lee
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  • Can we say anything similar about the case when metric is induced by embedding space? Also, can you please elaborate your statement and/or provide references/explain or give intuition why is this the case? – Vlad Apr 20 '17 at 22:59
  • @Vlad: https://en.wikipedia.org/wiki/Riemann_surface#Hyperbolic_Riemann_surfaces – Moishe Kohan Apr 20 '17 at 23:11
  • Reference : 167 p. Riemannian geometry - Gallot Hulin and Lafontaine – HK Lee Apr 20 '17 at 23:12
  • Are there simpler examples if I relax the condition to be only compact and negatively curved? – zach Apr 21 '17 at 02:06
  • I can not understand your question. only compact ? – HK Lee Apr 21 '17 at 02:08
  • Only compact and negatively curved but not necessarily of constant curvature. I am thinking of the poincare disk, but it is not compact. – zach Apr 21 '17 at 02:12
  • @zach The Pointcare disk has constant sectional curvatures. – ABIM Jan 06 '23 at 16:01