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.
32.
Let M _ i be a sequence of n dimensional Riemannian manifolds, where \ sec ( M _ i ) denotes the sectional curvature of the " i " th manifold.
33.
In two dimensions sectional curvature is always pointwise constant since there is only one two-dimensional subspace \ Pi _ p \ subset T _ p M, namely T _ p M.
34.
The 2 theorem states : a Dehn filling of " M " with each filling slope greater than 2 results in a 3-manifold with a complete metric of negative sectional curvature.
35.
Moreover, if the diameter is equal to ? / " " k ", then the manifold is isometric to a sphere of a constant sectional curvature " k ".
36.
If for each point in a connected Riemannian manifold ( of dimension three or greater ) the sectional curvature is independent of the tangent 2-plane, then the sectional curvature is in fact constant on the whole manifold.
37.
If for each point in a connected Riemannian manifold ( of dimension three or greater ) the sectional curvature is independent of the tangent 2-plane, then the sectional curvature is in fact constant on the whole manifold.
38.
On the other hand one can fix the bound of sectional curvature and get the diameter going to zero, so the almost-flat manifold is a special case of a collapsing manifold, which is collapsing along all directions.
39.
If tighter bounds on the sectional curvature are known, then this property generalizes to give a comparison theorem between geodesic triangles in " M " and those in a suitable simply connected space form; see Toponogov's theorem.
40.
Indeed, if \ xi is a vector of unit length on a Riemannian " n "-manifold, then is precisely times the average value of the sectional curvature, taken over all the 2-planes containing \ xi.
