This is the property of the following curvature being everywhere zero: sectional curvature
- The sectional curvature is identically zero, viz the sectional curvature is zero for every tangent plane at every point.
- The Riemann curvature tensor of the Levi-Civita connection is the zero map, viz vanishes everywhere
Relation with other properties
- Constant-curvature metric
- Ricci-flat metric
- Einstein metric
- Constant-scalar curvature metric
- Conformally flat metric
The restricted holonomy group of a flat metric is trivial. In other words, any loop homotopic to the constant loop, has trivial holonomy. Thus, the holonomy group is a homomorphic image of the fundamental group, and gives rise to a linear representation of the fundamental group.
Further information: Minding's theorem
This result says that two smooth surfaces with the same constant curvature are locally isometric.
The direct product of two Riemannian manifolds, both equipped with a flat metric, also gets the flat metric. Thus, since every curve is a flat manifold, any manifold obtained as a direct product of curves is also flat.