By uniformization theorem we can classify manifolds. In this theorem classification is took by constancy of sectional curvature. The manifolds with constant sectional curvature are called space forms. One could classfy the manifold by constant sectional curvature; positive, negative and zero.

I agree with Inan Unal, one can classify the manifold with respect to the values of sectional curvature tensors. But my quastion devoted to the kind of geometries which are determined by these values.

The answer is

1- Positve sectional curvature implies elliptic geometry

2- negative sectional curvature implies hyperbolic geometry

3- zero sectional curvature implies Euclidean geometry.

There are three kinds of geometry that detremined regarding the values of sectional curvature as follows:

1- Elleptic geometry: positive curvature +1

2- Euclidean geometry: zero curvature

3- Hyperbolic geometry: negative curvature -1