A consequence of this formula is that the sectional curvature satisfies 1 \ leq K ( \ sigma ) \ leq 4 for all 2-planes \ sigma.
23.
The standard counterexample is complex projective space with the Fubini Study metric; sectional curvatures of this metric take on values between 1 and 4, with endpoints included.
24.
Ricci curvature and more specifically those with negative sectional curvature . ( A strange and interesting fact is that all closed three-manifolds admit metrics with negative Ricci curvatures!
25.
This system is useful in physics, especially in the physical quantities are identified with geometric quantities such as areas, lengths, dimensionless numbers, path curvatures, or sectional curvatures.
26.
In particular, the four-dimensional manifold " S " 2 & times; " S " 2 should admit no Riemannian metric with positive sectional curvature.
27.
:Suppose is complete, connected and non-compact with sectional curvature, and there exists a point in where the sectional curvature ( in all sectional directions ) is strictly positive.
28.
:Suppose is complete, connected and non-compact with sectional curvature, and there exists a point in where the sectional curvature ( in all sectional directions ) is strictly positive.
29.
For example, Gromov and Lawson showed that a closed manifold that admits a metric with sectional curvature d " 0, such as a torus, has no metric with positive scalar curvature.
30.
Another consequence of the Geometrisation conjecture is that any closed 3-manifold which admits a Riemannian metric with negative sectional curvatures admits in fact a Riemannian metric with constant sectional curvature-1.
