Constant-scalar curvature 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 constant: scalar curvature
Definition
Symbol-free definition
A Riemannian metric on a differential manifold is said to be a constant-scalar curvature metric if the scalar curvature at all points is equal.
Definition with symbols
Fill this in later
Relation with other properties
Stronger properties
The following properties of Riemannian metrics are stronger than the property of being a constant-scalar curvature metric:
- Flat metric
- Conformally flat metric
- Constant-curvature metric
- Einstein metric
- Ricci-flat metric
- Homogeneous metric
- Locally homogeneous metric
- Zero-scalar curvature metric
Weaker properties
Flows
Yamabe flow
The Riemannian metrics with this property are precisely the stationary points for this flow: Yamabe flow
The Yamabe conjecture states that every conformal class of Riemannian metrics contains a constant-scalar curvature metric. The Yamabe flow is intended to help answer this conjecture positively. The Yamabe flow has, for its stationary points, the metrics of constant scalar curvature. Further, the Yamabe flow preserves the conformal class of the Riemannian metric.
The idea is to start with an arbitrary member of the conformal class, evolve it using the Yamabe flow, and take the limit as .
