---
date: 2022-01-15 21:49
modification date: Friday 11th February 2022 23:10:49
title: curvature
aliases: [curvature, curvature of a connection, p curvature, curvature form, parallel, geodesic, geodesics, parallel transport]

---

Tags: 
#geomtop/Riemannian-geometry #physics #geomtop/differential-geometry 
Refs:
[Riemannian Geometry](Unsorted/Riemannian%20Geometry.md)
[curvature](Unsorted/curvature.md)
[Lie algebra valued form](Lie%20algebra%20valued%20form.md)

# curvature


# Definition

Differential geometry, in terms of the [covariant derivative](Unsorted/covariant%20derivative.md):
![](attachments/2023-03-10diffgeocurv.png)

Given a [Riemannian Geometry](Unsorted/Riemannian%20Geometry.md) $\nabla$ on $E\searrow M$ a [Riemannian manifold](Unsorted/Riemannian%20manifold.md) the **curvature form** is a 2-form $F_\nabla$ with values in $\Endo(E) \cong E\dual \tensor E$, i.e.
$$
F_\nabla \in \Omega^2_{M}(\Endo E) \cong \globsec{\Extalg^2 \TM \tensor \Endo E} 
\qquad 
F_\nabla(X, Y)(\wait) = \qty{ [\nabla_X, \nabla_Y] -\nabla_{[X, Y]}} (\wait)
$$

The [covariant exterior derivative](Unsorted/covariant%20derivative.md) satisifes $d_\nabla^2 s = F_\nabla \wedge s$ for $s\in \drcomplex_M(E)$ an $E\dash$valued form, and thus $d^2=0$ when the curvature form vanishes.
This yields a [flat bundle](Unsorted/flat%20bundle.md).


![](attachments/Pasted%20image%2020220408005556.png)

## Cartan's Structure Equation

If $\nabla = A$ locally as a matrix of 1-forms, then $F_\nabla = dA + A\wedgepower{2}$ or $F_\nabla = dA + {1\over 2}[A, A]$.

# Other types

Types of curvature:

- [Riemannian curvature](Unsorted/Riemannian%20curvature.md)
	- [Ricci curvature](Unsorted/Ricci%20curvature.md): contract the Riemann curvature
	- [scalar curvature](Unsorted/scalar%20curvature.md): take the trace of Ricci curvature
- [Gaussian curvature](Gaussian%20curvature)
	- [Sectional curvature](Sectional%20curvature): defined in terms of the Riemannian curvature, and determines the Riemannian curvature completely.
	Essentially the Gaussian curvature of the geodesic surface with the same tangent at $p$, i.e. the image of $\exp_p$.

How these are related: recover [Ricci curvature](Ricci%20curvature.md) by contracting [Unsorted/Riemannian curvature](Unsorted/Riemannian%20curvature.md), and you get [scalar curvature](scalar%20curvature.md) by taking the trace of [Ricci curvature](Ricci%20curvature.md).


## For 3-manifolds?

![](attachments/Pasted%20image%2020220501002751.png)

# Parallel vector fields and geodesics

- Can define **parallel** vector fields as $\nabla X = 0$, a PDE.
	- Don't generally exist, this is an overdetermined equation.
	The integrability condition for this equation is equivalent to $\Curv(\nabla) = 0$.
	- If curvature vanishes, parallel transport along every curve can be used to define parallel vector fields on $M$.

![](attachments/2023-03-10parallel.png)
![](attachments/2023-03-10geodesics.png)

## Parallel transport

![](attachments/2023-03-10paralleltransport.png)

# p Curvature

![Pasted image 20211122004915.png](Pasted%20image%2020211122004915.png)

![](attachments/Pasted%20image%2020220218024215.png)