[This very rough outline is just a placeholder. I have been putting off writing this article, until I can draw a good diagram.]
“Recall” a connection 
instructs how to compare vectors at nearby points on a manifold. It is essential for defining what “parallel” means, and taking certain derivatives. In physics, it is typically obtained from a metric. This case is called a metric connection or Levi-Civita connection. The most familiar expression of a connection is the connection coefficients 
, which are called Christoffel symbols in the special case of a metric connection.
One intuition is the gradient of basis vector (fields). 
. This decomposes the gradient of one basis vector field in terms of all the basis vectors. In fact, this gradient is expressed in totality by the total covariant derivative 
… For each fixed value of 
(and choice of basis), this is actually a tensor, of rank 2.
However an even better one is a map from vectors to rotations. Input vector is the direction a frame is moved along the manifold. (To be slightly more precise, for an “infinitesimal” distance. Though one could certainly be much clearer still.) The output is the rotation the frame undergoes. Relative to the frame (field) already defined there, that is.
Cartan knew this interpretation. For example his book Riemannian geometry in an orthogonal frame 
is clear. It is not incidental, but infused throughout. Starting with section 1 of chapter 1! I am thoroughly impressed.(While I was well aware Cartan is a famous geometer, naturally I have not read much of his work carefully in the original sources. firsthand, experiential.)
I am deeply passionate about geometric intuition, because I like deep conceptual understanding. Conversely I feel very unsatisfied when equations are presented without meaning behind them — especially things like connections or curvature tensors, in a pedagogical setting. (It must be hard to teach a course on general relativity, where even the requisite differential geometry must be crammed in first. Besides it may not even be a lecturer’s specialty. Hence I am not criticising courses… much. But even most textbooks don’t present the degree of intuition which I would prefer. But at least one piece of good news for me personally is that this gives me an avenue where I can contribute. And I feel enthusiastic about that. Now this won’t actually be original understanding, in many cases. It would be more about drawing attention to under-represented insights.)
It seems intuition sometimes gets lost, historically. One cause seems to be abstraction and generalisation. Now, I have absolutely no problem with those. It is valuable and insightful, and it is good that specialists work on that. But I like to also see clear motivation for introducing a quantity. And knowing the original historical motives by the discoverer(s) can make it more relatable. Covectors are a good example. Even if a strange circuitous route (like a recent video about optics and Lagrangians iirc), because connects to existing insights. Also could add confidence: “I could do that too!” [The opposite is refining and polishing results. Now that is certainly valuable, in fact it is absolutely essential (pragmatically speaking) because it is too much work to read original sources, in most cases. But this already receives tremendous emphasis.]
from the connection in the usual sense). I will write more on this in the future.
evaluates in detail. I certainly appreciate the modern curvature-only view for what it affirms, just not what it denies.
. In our universe, matter forms a “cosmic web”, which includes filaments with a heavy “node” at the end — a cluster or supercluster. On the other hand, cosmic “voids” are vast regions containing less matter than average. Over time, the matter clumps further, while the voids grow larger and sparser.
. This is normally understood as an “internal symmetry”. But Hestenes has a remarkable proposal to interpret it geometrically in spacetime itself, using tangent vectors and bivectors formed from them, in a 
. So we seek an analogue of this structure in Minkowski spacetime. Choose an orthonormal basis: 
etc. Hestenes defines the bivectors 
, 
and 
. This notation uses the geometric product, but in this case the vectors are orthogonal, so the result is just wedge products, e.g. 
. Together, the scalar and pseudoscalar parts from the geometric algebra form a subalgebra which is isomorphic to 
, the underlying field for the vector space 
(or maybe some subgroup), hence these spacetime “rotations” include both boosts and spatial rotations. Then 
encodes a boost in the 
-direction, combined with a rotation about the 
, the rotations on a 6-dimensional real space. Specifically, those which preserve the structure of the complex axes, relative to one another. However in our case these are not spacetime rotations, but act on the space of bivectors. This is quite abstract, and I envision making illustrations in future work. But just as a 2D rotation changes the 
-components of a vector, these 
and so on). Returning to 
R
G
R
G
R
\mathfrak{su}(3)
SU(3)
SU(3)
\mathfrak{su}(3)
\mathfrak{su}(3)
\epsilon
R\wedge_{\mathbb C}G\wedge_{\mathbb C}B$ is valid. The wedge here is not the one acting on tangent vectors, but “complex vectors” which for us are basically bivectors. Intuitively, the “complex trivector” fills all directions of (real) bivector space. Just like the 4-volume element formed from vectors fills all spacetime directions (within a tangent space at a point, that is).
, the connection coefficients are:![\[ \Gamma_{\mu\nu}^{\hphantom{\mu\nu}\sigma} := \langle\nabla_\mu\mathbf e_\nu,\mathbf e^\sigma\rangle. \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-0da3696c8ef5490c24c4ced6dac4df8f_l3.png "Rendered by QuickLaTeX.com")
, as introduced in the 
of differentiation, the second is the basis vector (field) being differentiated, and the last index is for the component of the resulting vector. We use the same angle bracket notation for the metric scalar product, inverse metric scalar product, and the contraction of a vector and covector (as above; and this does not require a metric).
variant will be derived from that.
, as 
, whereas for the Greek index the metric components in a coordinate frame 
are used. However while these textbook(s) gave useful practical formulae, I did not find them clear on what was definition vs. what was derived. I eventually concluded the various indices and their placements are best treated as a definition of components, with any formulae for swapping / raising / lowering being obtained from that.]
, with all indices lowered. These quantities are the overlaps between the vectors 
and 
. 
. After all, for any vector, 
, which may be checked by contracting both sides with 
. ]
s, note:![\[ \Gamma_{\mu\nu\tau} = \langle\nabla_\mu\mathbf e_\nu,\mathbf e^\sigma g_{\sigma\tau}\rangle = \langle\nabla_\mu\mathbf e_\nu,\mathbf e^\sigma\rangle g_{\sigma\tau} = \Gamma_{\mu\nu}^{\hphantom{\mu\nu}\sigma}g_{\sigma\tau}, \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-1e8dc7263a128cfd70916ce655732f69_l3.png "Rendered by QuickLaTeX.com")
have been called “Christoffel symbols of the first kind”.
. However, this begs the question of what the notation ‘
’ means. The raised index suggests 
, which is a covector, but we can take its dual using the metric (assuming there is one). Hence 
is a sensible definition, or for an arbitrary covector, 
. (For those not familiar with the notation, 
is simply the vector with components 
.)
, where 
are still the inverse metric components for the original basis. Hence 
, by the linearity of this slot. In terms of the coefficients, 
, hence the first index is also “tensorial”.
. This means it is now a covector field which is being differentiated, and the components of the result are peeled off. For their relation to the original coefficients, note firstly 
, since these bases are dual. These are constants, hence their gradients vanish:![\[ 0 = \nabla_\mu \langle\mathbf e^\sigma,\mathbf e_\nu\rangle = \langle\nabla_\mu\mathbf e^\sigma,\mathbf e_\nu\rangle + \langle\mathbf e^\sigma,\nabla_\mu\mathbf e_\nu\rangle. \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-0f37b17e9f5fc89208a9b894a85bb478_l3.png "Rendered by QuickLaTeX.com")
. Note the order of the last two indices is swapped in this expression, and their “heights” are also changed. (I mean, it is not just that 
and 
.
using our conventions. Both equalities follow from conveniently chosen properties of bases: the dual basis relation in the former case, and orthonormality of a single basis in the latter. Our relation holds in any basis, and does not require a metric. But both relations are useful — they just apply to different contexts.
. Only now have we invoked the metric, in our discussion of the middle index.
-linear but not 
-linear.)
, for some functions 
with 3 × 2 = 6 indices.
, is similar. This requires a metric, and gives a unique interpretation to raising and lowering indices. But in fairness, most transformations we have described also require a metric.)![\[ \boldsymbol\omega_\mu^{\hphantom\mu\nu} = \omega_{\mu\hphantom\nu\tau}^{\hphantom\mu\nu}\mathbf e^\tau, \qquad\qquad \boldsymbol\omega^\nu_{\hphantom\nu\mu} = \omega^\nu_{\hphantom\nu\mu\tau}\mathbf e^\tau, \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-f397e1fbf79d43687701700aa42670d2_l3.png "Rendered by QuickLaTeX.com")
![\[ \nabla_\sigma\mathbf e_\mu = \langle\nabla_\sigma\mathbf e_\mu,\mathbf e^\tau\rangle\,\mathbf e_\tau = \omega_{\mu\hphantom\tau\sigma}^{\hphantom\mu\tau}\mathbf e_\tau. \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-2d5e1b1e197351d1ed7f9e679381e304_l3.png "Rendered by QuickLaTeX.com")
‘. But can check it holds by contracting both sides with 
. Similarly for the covector gradients,![\[ \nabla_\sigma\mathbf e^\nu = \langle\nabla_\sigma\mathbf e^\nu,\mathbf e_\tau\rangle\,\mathbf e^\tau = \omega^\nu_{\hphantom\nu\tau\sigma}\mathbf e^\tau = -\omega_{\tau\hphantom\nu\sigma}^{\hphantom\tau\nu}\mathbf e^\tau. \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-21127b0f5cf22f54c65756c7ef1ccf83_l3.png "Rendered by QuickLaTeX.com")
. Because: substitute 
into the left slot (in our convention) of both sides. The RHS becomes: 
, by linearity of this slot. Now, apply this identity to ![\[ \nabla\mathbf e_\mu = \mathbf e^\sigma\otimes\nabla_\sigma\mathbf e_\mu = \omega_{\mu\hphantom\tau\sigma}^{\hphantom\mu\tau}\mathbf e^\sigma\otimes\mathbf e_\tau = \boldsymbol\omega_\mu^{\hphantom\mu\tau}\otimes\mathbf e_\tau. \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-8f2df818e811d8acf1ea5b7f97bb442a_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{split} \nabla\mathbf e^\nu &= e^\sigma\otimes\nabla_\sigma\mathbf e^\nu = \omega^\nu_{\hphantom\nu\tau\sigma}\mathbf e^\sigma\otimes\mathbf e^\tau = -\omega_{\tau\hphantom\nu\sigma}^{\hphantom\tau\nu}\mathbf e^\sigma\otimes\mathbf e^\tau \\ &= \boldsymbol\omega^\nu_{\hphantom\nu\tau}\otimes\mathbf e^\tau = -\boldsymbol\omega_\tau^{\hphantom\tau\nu}\otimes\mathbf e^\tau. \end{split}\]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-9faf56c751b9f2cc0c5b5ad6fee9bd9e_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{split} d(\mathbf e^\nu) &= \boldsymbol\omega^\nu_{\hphantom\nu\sigma}\wedge\mathbf e^\sigma = \omega^\nu_{\hphantom\nu\sigma\tau}\mathbf e^\tau\wedge\mathbf e^\sigma = -\omega^\nu_{\hphantom\nu\sigma\tau}\mathbf e^\sigma\wedge\mathbf e^\tau \\ &= \omega_{\sigma\hphantom\nu\tau}^{\hphantom\sigma\nu}\mathbf e^\sigma\wedge\mathbf e^\tau. \end{split}\]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-b93a68da7af62887c0b6ec1cf1a7a72c_l3.png "Rendered by QuickLaTeX.com")
are one way to express a connection 
are more familiar and achieve the same purpose, but package the information differently. Connection forms are part of Cartan’s efficient and elegant “moving frames” approach to derivatives and curvature.
, where 
for the contraction between a vector and covector, which is not uncommon in the literature. The unified notation with the metric scalar product is convenient, although it is sometimes worth reminding oneself that no metric is needed in this particular case.) To find the components, substitute basis vectors 
:![\[ \omega_{\mu\hphantom\nu\sigma}^{\hphantom\mu\nu} := \boldsymbol\omega_\mu^{\hphantom\mu\nu}(\mathbf e_\sigma) = \langle\nabla_\sigma\mathbf e_\mu,\mathbf e^\nu\rangle =: \Gamma_{\sigma\mu}^{\hphantom{\sigma\mu}\nu}, \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-d5fddf068b3b2393d8df910a2ab6e05f_l3.png "Rendered by QuickLaTeX.com")
as usual. Hence with our conventions, the 
-index specifies which basis vector field is being differentiated, 
for our expression — which swaps the index order.)![\[ \omega\indices^\nu_{\hphantom\nu\mu\sigma} := \boldsymbol\omega^\nu_{\hphantom\nu\mu}(\mathbf e_\sigma) := \langle\nabla_\sigma\mathbf e^\nu,\mathbf e_\mu\rangle = -\langle\nabla_\sigma\mathbf e_\mu,\mathbf e^\nu\rangle = -\omega_{\mu\hphantom\nu\sigma}^{\hphantom\mu\nu}. \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-27ef19a830d62d9e98a93fea6871076a_l3.png "Rendered by QuickLaTeX.com")
![\[ \boldsymbol\omega_\mu^{\hphantom\mu\nu} = -\boldsymbol\omega^\nu_{\hphantom\nu\mu}. \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-401382997b6ad9a63a20ec0bfb60ae95_l3.png "Rendered by QuickLaTeX.com")
![\[ 0 = \nabla_\sigma\langle\mathbf e_\mu,\mathbf e^\nu\rangle = \langle\nabla_\sigma\mathbf e_\mu,\mathbf e^\nu\rangle + \langle\mathbf e_\mu,\nabla_\sigma\mathbf e^\nu\rangle. \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-832cdb5ac5b43f58a1f9bede19948998_l3.png "Rendered by QuickLaTeX.com")
is constant, so its gradient vanishes. The second equality follows from the defining properties of the covariant derivative, i.e. the extension of the connection to covectors and other tensors (e.g. Lee 2018 
. The Latin index is raised and lowered using the metric components in the arbitrary frame, whereas the Greek index uses the metric components in the coordinate frame. However textbooks were not clear on what was definition vs. what was derived, I thought. I eventually concluded the various indices and their placements are best treated as a definition of components, with any formulae for swapping/raising/lowering being obtained from that.]
, in terms of our notation. I did not see how to prove this, so initially I just copied and affirmed it. But I am now updating this 6 months later, and I realise it is only true for a metric of Riemannian signature. Sorry about that.
leads to 
. Multiply both sides by 
and sum, which gives:
. On the LHS, this raised the second index, which is tensorial or “linear”. But on the RHS, the first index doesn’t obey the tensorial rule. (See the next article I would write on 
. Now apply orthonormality again, so the sum collapses to 
, hence:![\[ \boldsymbol\omega_\mu^{\hphantom\mu\nu} = -\eta_{\mu\mu}\eta_{\nu\nu}\,\boldsymbol\omega_\nu^{\hphantom\nu\mu}. \]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-4cfe1bdd90f00e5b6ca2a839c19a0fab_l3.png "Rendered by QuickLaTeX.com")
.
, whereas a covector has a lowered index: 
. Similarly the dual to 
, using the metric, is the covector with components denoted 
say; while the dual to 
is the vector with components 
.
. Similarly, the component notation 
. These bases are taken to be dual to one another (in the sense of bases), meaning: 
. We also have 
and 
, where as usual 
and 
. (The “sharp” and “flat” symbols are just a fancy way to denote the dual. This is called the musical isomorphism.)
. This gives a different decomposition than in the previous paragraph. Curiously, this expression contains a raised component index, even though it describes a covector. For each index value 
is a different decomposition of the vector 
. It describes a vector, despite using a lowered component index. Using the metric, the two vector bases are related by: 
.
. (While 
. In the diagram we index the vectors by colour, rather than the typical 1, 2, and 3. Rather than using a co-basis of covectors as in the standard modern approach, we take both bases to be made up of vectors. Note for example, the red vector on the right is orthogonal to the blue and green vectors on the left. (I made up the directions when plotting, so don’t expect everything to be quantifiably correct.) Arguably “inverse basis” would be the best name. Finally I would prefer to start from this picture, as it is simple and intuitive, rather than start with covectors and musical isomorphisms. Maybe next time I will be brave enough.
, which describes an orthogonal matrix. While admittedly the paper is mostly abstract algebra, he is also motivated by geometry. In §3 he mentions that the equation for a surface is “transformed” under a change of [Cartesian] coordinates, including the case where the coordinate origins coincide. We recognise this (today, at least) as a rotation, possibly combined with a reflection. Euler also mentions “angles” (§4 and later), which is clearly geometric language.![\[\begin{tabular}{|c|c|c|}\hline $\frac{p^2+q^2+r^2+s^2}{u}$ & $\frac{2(qr+ps)}{u}$ & $\frac{2(qs-pr)}{u}$ \\ \hline $\frac{2(qr-ps)}{u}$ & $\frac{p^2-q^2+r^2-s^2}{u}$ & $\frac{2(pq+rs)}{u}$ \\ \hline $\frac{2(qs+pr)}{u}$ & $\frac{2(rs-pq)}{u}$ & $\frac{p^2-q^2-r^2+s^2}{u}$ \\ \hline\end{tabular},\]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-8a820b1a5889a514cf73584ad468f584_l3.png "Rendered by QuickLaTeX.com")
. (I have copied the style which Euler uses in some subsequent examples.) By choosing rational values of the parameters, the matrix entries will also be rational, however this is not our concern here. The matrix has determinant +1, so we know it represents a rotation. It turns out the parameters form the components of a spinor!! 
are the real components of a normalised spinor. We allow all real values, but will ignore some trivial cases. One aspect of spinors is clear from inspection: in the matrix the parameters occur only in pairs, hence the sets of values 
give rise to the same rotation matrix. (Those familiar with spinors will recall the spin group is the “double cover” of the rotation group.)![\[q\hat{\mathbf y}\wedge\hat{\mathbf z} + r\hat{\mathbf z}\wedge\hat{\mathbf x} + s\hat{\mathbf x}\wedge\hat{\mathbf y}.\]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-46fe75d65665b9ede3e4bcd9bc9e579a_l3.png "Rendered by QuickLaTeX.com")
for example, as the parallelogram or plane spanned by those vectors. It also has a magnitude and handedness/orientation.) The sum is itself a plane, because we are in 3D. Dividing by 
gives a unit bivector. For the spinor, the angle of rotation θ/2 say, is given by (c.f. Doran & Lasenby 2003 ![\[\cos(\theta/2) = p/\sqrt{u},\qquad \sin(\theta/2) = \sqrt{(q^2+r^2+s^2)/u}.\]](http://cmaclaurin.com/cosmos/wp-content/ql-cache/quicklatex.com-3129e321d8198223313e27e17b0297f0_l3.png "Rendered by QuickLaTeX.com")
, where I label by u the quantity 
mentioned by Euler, might form spinors of 4D space (not 3+1-dimensional spacetime). If so, these are members of Spin(4), the double cover of the 4D rotation group SO(4).
