Does there exist a known example of Riemannian manifold who its sectional curvature admit both zero and positive values?

Question

Does there exist a known example of Riemannian manifold who its sectional curvature admit both zero and positive values ?

Dietrich Burde · Accepted Answer · 2017-11-15T15:00:50.357

4

For $2$-manifolds, the sectional curvature is the Gauss curvature, see

Sectional Curvature, Gauss curvature

Theorem (From Do Carmo's book, p. 282, see below): Let $S\subseteq \mathbb{R}^3$ be a connected, regular, compact, orientable surface which is not homeomorphic to a sphere. Then there are points on $S$ where the Gaussian curvature is positive, negative, and zero.

Proof: Compact surface with Gaussian curvature is positive, negative, and zero

See here for a nice example on the $2$-torus.

edited Nov 15 '17 at 15:00

answered Nov 15 '17 at 13:27

Dietrich Burde

130,978

Thanks, Is this theorem true is higher dimensions? – C.F.G Nov 15 '17 at 14:13
1

Well, you could just take direct products, e.g. $T^2 \times \mathbb{R}^n$ to obtain higher-dimensional examples. – Dietrich Burde Nov 16 '17 at 11:42

Does there exist a known example of Riemannian manifold who its sectional curvature admit both zero and positive values?

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