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title: Notes on Needham's Derivation of the Riemann Curvature Tensor
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Needham defines the covariant derivative of a vector field $w$ in the direction of a vector (field) $\mathbf{v}$ as 

## Covariant Derivative Definition

$$\nabla_{\mathbf{v}} \mathbf{w} \asymp \frac{\mathbf{w}_{||}(q \to p) - \mathbf{w}(p)}{\epsilon}$$

where $q$ is a point "infinitesimally" close to $p$ in the direction of $\mathbf{v}$ and $\mathbf{w}_{||}(q \to p)$ is the parallel transport of $\mathbf{w}(q)$ back to $p$ along the geodesic from $q$ to $p$.  Needham also indicates this derivative represents how much the vector $\mathbf{w}(q)$ differs from its original position $\mathbf{w}(p)$ after being moved back to $p$ via parallel transport.  It is implied, then, that the point at which this derivative is evaluated is $p$

$$\nabla_{\mathbf{v}}(p) \mathbf{w} \asymp \frac{\mathbf{w}_{||}(q \to p) - \mathbf{w}(p)}{\epsilon}.$$

## The Vector Holonomy

Recall, the holonomy $$R_U(K) \equiv \delta_K(\angle \mathbf{w}_{||})$$ measures the angle **FROM** a fixed fiducial vector field $U$ - **TO** the parallel transported vector $\mathbf{w}_{||}$ along $K$.  If we choose the fiducial vector field $U$ to be the vector field $w$ itself, the holonomy measures the angle from $w$ to its parallel transport $\mathbf{w}_{||}$ along the infinitesimal curve $K$ from $p$ to $q$.  This is exactly what the covariant derivative measures, except for a negative sign.  


## The Negative Holonomy 

The negative vector holonomy $-R_U(K)$ measures the opposite - the angle **FROM** the parallel transported $\mathbf{w}_{||}$ - **TO** the fiducial vector field.  This is exactly what the covariant derivative measures in the infinitesimal case!  That is,    
$$\nabla_{\mathbf{v}} \mathbf{w} \asymp \frac{\mathbf{w}_{||}(q \to p) - \mathbf{w}(p)}{\epsilon} -R_{\mathbf{w}}K).$$

## Computing Holonomy Along the 5 Segments

By the Mean Value Theorem, the negative vector holonomy along $K$ (being a continuous, differentiable function measuing the change in $w$ from the vector field - to the PT'd vector along $K$) is ultimatly 
$$-\delta_{oa}(\angle \mathbf{w}_{||} w) \asymp \mathbf{w}(o^*) - \mathbf{w}_{||}(o^*) \asymp \nabla_{\mathbf{v}} \mathbf{w}(o^*) $$ where $K = oa$ is in the direction of $\mathbf{v}$ and $o^*$ lies between $o$ and $a$. 

The full negative holonomy of the loop can be computed as the sum of the negative vector holonomies along each of the 5 segments that make up $K$. 

## Putting it all Together
Thus, the covariant derivative can be computed as the sum of the negative vector holonomies along each segment of $K$:
$$ -R_{\mathbf{w}}(L) \asymp -R_{\mathbf{w}}(oa) - R_{\mathbf{w}}(ab) - R_{\mathbf{w}}(bq) - R_{\mathbf{w}}(qp) - R_{\mathbf{w}}(po) $$
$$ \asymp \nabla_{\delta \mathbf{u} \, u} \mathbf{w}(o^*) + \nabla_{\delta \mathbf{v} \, \mathbf{v}} \mathbf{w}(a^*) + \nabla_c \mathbf{w}(c^*) - \nabla_{\delta \mathbf{u} \, u} \mathbf{w}(q^*) - \nabla_{\delta \mathbf{v} \, \mathbf{v}} \mathbf{w}(p^*) $$

where $a^*, b^*, c^*, p^*, q^*$ are points along each segment of $K$ coming from the Mean Value Theorem.  The $p^*$ point lies between $p$ and $o$, so we can approximate it as the midpoint of $op$.  Similarly for the other $^*$ points, $q^*$ is approximately the midpoint of $pq$, $c^*$ approximately the midpoint of $bq$ and $a^*$ approximately the midpoint of $ab$.  

The $p^*$ point lies in a segment traversed in the $-v$ direction, so we have a negative sign in front of that term.  Similarly for the $q^*$ point.  Thus, we have
$$ -R_{\mathbf{w}}(L) \asymp \delta \mathbf{u} \nabla_{\mathbf{u}} \mathbf{w}(a^*) + \delta \mathbf{v} \nabla_{\mathbf{v}} \mathbf{w}(b^*) + \nabla_c \mathbf{w}(c^*) -\delta \mathbf{u} \nabla_{\mathbf{u}} \mathbf{w}(q^*) -\delta \mathbf{v} \nabla_{\mathbf{v}} \mathbf{w}(p^*) $$

we prepare to collect the terms with $\delta \mathbf{u}$ and $\delta \mathbf{v}$ we collect like terms with color coding:

$$ -R_{\mathbf{w}}(L) \asymp
\tcboxmath[size=small,colback=red!30]{\delta \mathbf{u} \nabla_{\mathbf{u}} \mathbf{w}(a^*)} 
\tcboxmath[size=small,colback=green!30]{+ \delta \mathbf{v} \nabla_{\mathbf{v}} \mathbf{w}(b^*)}
\tcboxmath[size=small,colback=blue!30]{+ \nabla_c \mathbf{w}(c^*)}
\tcboxmath[size=small,colback=red!30]{-\delta \mathbf{u} \nabla_{\mathbf{u}} \mathbf{w}(q^*)} 
\tcboxmath[size=small,colback=green!30]{+\delta \mathbf{v} \nabla_{\mathbf{v}} \mathbf{w}(p^*)}
$$


Notice that $a^*$ and $q^*$ are both approximately the midpoint of $ab$ and $pq$, respectively, so we can combine those terms.  Similarly for $b^*$ and $p^*$.  This gives us

$$ -R_{\mathbf{w}}(L) \asymp  
\tcboxmath[size=small,colback=green!30]{
\delta \mathbf{v} \left( \nabla_{\mathbf{v}} \mathbf{w}(b^*) 
- \nabla_{\mathbf{v}} \mathbf{w}(p^*) \right)
}
- 
\tcboxmath[size=small,colback=red!30]{
\delta \mathbf{u} \left( \nabla_{\mathbf{u}} \mathbf{w}(q^*) 
- \nabla_{\mathbf{u}} \mathbf{w}(a^*) \right)  }
+ \tcboxmath[size=small,colback=blue!30]{\nabla_c \mathbf{w}(c^*)}
$$

Recalling the vector $c$ is the scaled commutator $$c \asymp \delta \mathbf{u} \, \delta \mathbf{v} [\mathbf{v},\mathbf{u}]  \asymp -\delta \mathbf{u} \, \delta \mathbf{v} [\mathbf{u},\mathbf{v}]$$ so we have

$$ -R_{\mathbf{w}}(L) \asymp  
\tcboxmath[size=small,colback=green!30]{
\delta \mathbf{v} \left( \nabla_{\mathbf{v}} \mathbf{w}(b^*) 
- \nabla_{\mathbf{v}} \mathbf{w}(p^*) \right)
}
- 
\tcboxmath[size=small,colback=red!30]{
\delta \mathbf{u} \left( \nabla_{\mathbf{u}} \mathbf{w}(q^*) 
- \nabla_{\mathbf{u}} \mathbf{w}(a^*) \right)  }
- \tcboxmath[size=small,colback=blue!30]{\nabla_{[\mathbf{u},\mathbf{v}]} \mathbf{w}(c^*)}
$$




$$ -R_{\mathbf{w}}(L) \asymp \delta \mathbf{v} 
\boxed{
\left( \nabla_{\mathbf{v}} \mathbf{w}(b^*) - \nabla_{\mathbf{v}} \mathbf{w}(p^*) \right)
}
+ \delta \mathbf{u} 
\boxed{
\left( \nabla_{\mathbf{u}} \mathbf{w}(q^*) - \nabla_{\mathbf{u}} \mathbf{w}(a^*) \right)
}
- 
\delta \mathbf{v} \, \delta \mathbf{u} \nabla_{[\mathbf{u},\mathbf{v}]} \mathbf{w}(c^*). $$

Applying the Mean Value Theorem again to the expressions boxed above, we get
$$ -R_{\mathbf{w}}(L) \asymp \delta \mathbf{u} \, \delta \mathbf{v} \nabla_{\mathbf{u}} \nabla_{\mathbf{v}} \mathbf{w}(r^*) - \delta \mathbf{v} \, \delta \mathbf{u} \nabla_{\mathbf{v}} \nabla_{\mathbf{u}} \mathbf{w}(s^*) - \delta \mathbf{u} \, \delta \mathbf{v} \nabla_{[\mathbf{u},\mathbf{v}]} \mathbf{w}(c^*) $$
where $r^*$ is between $a^*$ and $q^*$ and $s^*$ is between $b^*$ and $p^*$.  Both of these points are approximately the center of the loop $L$, so we can combine them into a single point $l^*$ at the center of $L$.  The point $c^*$ too becomes arbitrarily close to $l^*$ in the limit thus 

$$ -R_{\mathbf{w}}(L) \asymp \delta \mathbf{u} \, \delta \mathbf{v} \left( \nabla_{\mathbf{u}} \nabla_{\mathbf{v}} \mathbf{w}(l^*) - \nabla_{\mathbf{v}} \nabla_{\mathbf{u}} \mathbf{w}(l^*) - \nabla_{[\mathbf{u},\mathbf{v}]} \mathbf{w}(l^*) \right). $$   

Finally, dividing both sides by the area of the loop $\delta \mathbf{u} \, \delta \mathbf{v}$ taking the limit as the area goes to zero, we see $l^* \to o$ and 
$$ \lim_{L \to o} \frac{-R_{\mathbf{w}}(L)}{\text{Area}(L)} = \nabla_{\mathbf{u}} \nabla_{\mathbf{v}} \mathbf{w} - \nabla_{\mathbf{v}} \nabla_{\mathbf{u}} \mathbf{w} - \nabla_{[\mathbf{u},\mathbf{v}]} \mathbf{w} .$$

We can write this more compactly using the commutator notation, $$[u,v] = uv - vu$$ as $u= \nabla_{\mathbf{u}}$ and $v = \nabla_{\mathbf{v}}$ and reverting back to ultimate equality, and defining the Riemann Curvature Tensor as:
$$\boxed{ \mathcal{R}(\mathbf{u},\mathbf{v})\mathbf{w} \asymp  \frac{-R_{\mathbf{w}}(L)}{\text{Area}(L)} \asymp [\nabla_{\mathbf{u}}, \nabla_{\mathbf{v}}] \mathbf{w} - \nabla_{[\mathbf{u},\mathbf{v}]} \mathbf{w} .}$$    

