# nLab scalar curvature

### Context

#### Riemannian geometry

Riemannian geometry

## Basic definitions

• Riemannian manifold

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

• geodesic

• Levi-Civita connection

• ## Theorems

• Poincaré conjecture-theorem
• ## Applications

• gravity

• # Contents

## Definition

For $(X,e)$ a (pseudo-)Riemannian manifold with smooth manifold $X$ and vielbein field $e$, its scalar curvature is the smooth function

$R(e) \in C^\infty(X, \mathbb{R})$

defined to be the trace of the Ricci tensor of $e$

$R(e) \coloneqq tr_e Ric(c) \,.$

Last revised on April 25, 2018 at 03:17:06. See the history of this page for a list of all contributions to it.