Flat metric

This article defines a property that makes sense for a Riemannian metric over a differential manifold

This is the property of the following curvature being everywhere zero: sectional curvature

Definition

Symbol-free definition

A Riemannian metric on a differential manifold is said to be a flat metric if it satisfies the following equivalent conditions:

• 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

Weaker properties

• Constant-curvature metric
• Ricci-flat metric
• Einstein metric
• Constant-scalar curvature metric

Facts

Holonomy

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.

Minding's theorem

Further information: Minding's theorem

This result says that two smooth surfaces with the same constant curvature are locally isometric.

Metaproperties

Direct product-closedness

This property of a Riemannian metric on a differential manifold is closed under taking direct products of manifolds

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.