Riemann curvature tensor

Motivation

(see my own answer in math.stackexchange)

In a surface we compute holonomy by parallel transporting a vector around a loop $L$ and measuring the angle change. This angle does not depends on the chosen vector to be transported. When we shrink the loop we obtain Gaussian curvature as the holonomy per unit area.

Given a pseudo-Riemannian manifold $(M,g)$,we can consider the induced parallel transport and imitate the construction above. For example, we can try with a parallelogram $L$, instead of a general loop, generated by two vectors $u$ and $v$. (Keep an eye: we are not explaining how to construct that parallelogram)
A first problem arises by the fact that the parallelogram fails to close (it has to do with Lie bracket). But suppose we know how to close the gap.
The other difference is that we can choose different vectors to transport, obtaining different values. In @needham2021visual it is defined the vector holonomy $\delta w_{\parallel }$ as the difference between the original vector $w_0$ and the parallel transported vector $w_{\parallel }$ along the "quasi-parallelogram" $L$. The length of this vector is the Riemann curvature operator.

I think it should be denoted $\mathcal{R}(\mathbf{u}\delta u,\mathbf{v}\delta v)(\mathbf{w}):=-\delta {\mathbf{w}}_{\parallel }$, since it is an operator.

We can "see" it like a generalization of Gaussian curvature, since as $w$ is a unit vector, $\delta \mathrm{\Theta }=|\delta w_{\parallel }|$ (a "scalar" holonomy like for surfaces), so

See @needham2021visual page 287.

It can be shown (@needham2021visual page 290), that this ideas justifies the following

Definition

@lee2006riemannian
Given an affine manifold $(\mathcal{M},\mathcal{O},\mathcal{A},\mathrm{\nabla })$, the Riemann curvature endomorphism is the map

defined by

denoted in @needham2021visual by

Indeed, it can be proven that a multilinear map (indeed, a tensor)

Also, it is called the Riemann curvature tensor to the (1-3)-tensor

From here we can obtain the sectional curvatures.

To think of the relation of this definition with the picture/idea above see @needham2021visual page 289.

In a local chart

Given a manifold $M$ with the Levi-Civita connection whose Christoffel symbols in a specific chart are denoted by ${\mathrm{\Gamma }}_{ij}^k$, the components of the Riemann curvature tensor are the functions $R_{jab}^i$ such that

or, in other words

They can be obtained from these Christoffel symbols via the following expression:

On the other hand, observe that since

in the given chart $(U,x)$ we have (taking ${\mathrm{\nabla }}_a:={\mathrm{\nabla }}_{{\partial }_{x_a}}$):

since $[{\partial }_{x_a},{\partial }_{x_b}]=0$. So the curvature tensor is a measure of the commutativity of partial derivatives of vector fields (in a flat manifold we would have Clairaut's theorem).

Relation to curvature 2-forms

This is the Cartan's second structural equation.

Relation to Gaussian curvature of surfaces

In a 2-dimensional Riemannian manifold $(S,g)$, we consider an orthonormal frame $e_1,e_2$ for the surface $S$ so $\mathrm{\Theta }$ is antisymmetric and the curvature form is $\mathrm{\Omega }=d\mathrm{\Theta }$. According to Cartan's second structural equation,

On the other hand, according to this we have that

is the Gaussian curvature, so $K=g(-R(e_1,e_2;e_1),e_2)$.

So suppose $X=x^i{e}_i$, $Y=y^i{e}_i$, $Z=z^i{e}_i$. Then

If we denote $Z^{\perp }=z^2{e}_1-z^1{e}_2$

This formula can be rewritten as

where $g(X,Z)$ and $g(Y,Z)$ are the inner products of the vectors $X,Z$ and $Y,Z$ respectively. In effect,

Note: in @lee2006riemannian,
Lemma 8.7. The Gaussian curvature of a Riemannian 2-manifold is related to the curvature tensor by the formula