Notes on Needham’s Covariant Derivative

Needham defines the covariant derivative of a vector field \(w\) in the direction of a vector (field) \(v\) as

Covariant Derivative Definition

\[\nabla_v w \asymp \frac{w_{||}(q \to p) - w(p)}{\epsilon}\]

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

\[\nabla_v(p) w \asymp \frac{w_{||}(q \to p) - w(p)}{\epsilon}.\]

The Vector Holonomy

Recall, the holonomy \[R_U(K) \equiv \delta_K(\angle w_{||})\] measures the angle FROM a fixed fiducial vector field \(U\) - TO the parallel transported vector \(w_{||}\) 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 \(w_{||}\) 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 \(w_{||}\) - TO the fiducial vector field. This is exactly what the covariant derivative measures in the infinitesimal case! That is,
\[\nabla_v w \asymp \frac{w_{||}(q \to p) - w(p)}{\epsilon} -R_w(K).\]

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 \[-\delta_{oa}(\angle w_{||} w) \asymp w(o^*) - w_{||}(o^*) \asymp \nabla_v w(o^*) \] where \(K = oa\) is in the direction of \(v\) 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 \(K\): \[ -R_w(L) \asymp -R_w(oa) - R_w(ab) - R_w(bq) - R_w(qp) - R_w(po) \] \[ \asymp \nabla_{\delta u \, u} w(o^*) + \nabla_{\delta v \, v} w(a^*) + \nabla_c w(c^*) - \nabla_{\delta u \, u} w(q^*) - \nabla_{\delta v \, v} w(p^*) \]

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 \[ -R_w(L) \asymp \delta u \nabla_{u} w(a^*) + \delta v \nabla_{v} w(b^*) + \nabla_c w(c^*) -\delta u \nabla_{u} w(q^*) -\delta v \nabla_{v} w(p^*) \]

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 \[ -R_w(L) \asymp \delta u \left( \nabla_{u} w(a^*) - \nabla_{u} w(q^*) \right) + \delta v \left( \nabla_{v} w(b^*) - \nabla_{v} w(p^*) \right) - \nabla_c w(c^*) \]

The vector \(c\) is the scaled commutator \[c \asymp \delta u \, \delta v [u,v]\] so we have \[ -R_w(L) \asymp \delta u \left( \nabla_{u} w(a^*) - \nabla_{u} w(q^*) \right) + \delta v \left( \nabla_{v} w(b^*) - \nabla_{v} w(p^*) \right) - \delta u \, \delta v \nabla_{[u,v]} w(c^*). \]

Applying the Mean Value Theorem again to the first two big parentheses, we get \[ -R_w(L) \asymp \delta u \, \delta v \nabla_v \nabla_u w(r^*) - \delta u \, \delta v \nabla_u \nabla_v w(s^*) - \delta u \, \delta v \nabla_{[u,v]} w(c^*) \] 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 \[ -R_w(L) \asymp \delta u \, \delta v \left( \nabla_v \nabla_u w(l^*) - \nabla_u \nabla_v w(l^*) - \nabla_{[u,v]} w(l^*) \right). \]

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

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