3-form torsion and Cartan structural equations

Question

First, my level of math isn't very high as I come from the physics world. I am trying to understand the derivation of Cartan's 3-form torsion. I've read Robert Bryant's answer in this thread: Relating curvature and torsion of a connection to those of a curve and I was hoping he or others here will be able to assist.

I understand the idea of the 2-form torsion and how it is a twist in space-the difference between the right-Cartan connection matrix when indices are "switched" $[C^a_{b c}] - [C^a_{cb}] $ = [Affine torsion]. What I don't understand is why is the 3-form torsion = $[\sigma]$^$[d \sigma]$ where $[\sigma]$ is the inexact differentials of the mapping $[F][dy]=[\sigma]$. I'm familiar with the first structure equation and I tried to read Slebodzinski's exterior forms, but it only confuses me as he writes that $\Omega^x_\mu = d \omega ^x_\mu + \omega^x _l$ ^ $\omega^\lambda _\mu$ and then $\Omega^x_\mu$ is the curvature and $\Omega^x$ is torsion form ($\omega$ here correspongs to the right cartan connection matrix [C], as far as I know).

If you can explain the derivation of the 3-form torsion, or just refer me to a good source where I could read about it I would very much appreciate it.

M.

Answer 1

Well, I don't know what you mean by 'natural coordinates', so I might be entirely off, but here's a possible interpretation of what you are asking about:

First, assume given an $n$-manifold $M$, a Riemannian metric $g$, and a $g$-compatible connection $\nabla$.

On an open subset $U\subset M$, suppose that one has a $g$-orthonormal frame field $X = (X_1,\ldots, X_n)$. Let $\sigma = (\sigma_1,\ldots,\sigma_n)$ be the dual coframing. Then there exist unique $1$-forms $\omega_{ij}=-\omega_{ji}$ that represent the coefficients of the connection $\nabla$ with respect to the frame field $X$. One then has the first structure equation $$ \mathrm{d}\sigma_i = -\omega_{ij}\wedge\sigma_j + \tfrac12T_{ijk}\,\sigma_j\wedge\sigma_k $$ where $T_{ikj}=-T_{ijk}$. Then the expression $$ \Psi = \tfrac16 T_{ijk}\, \sigma_i\wedge\sigma_j\wedge\sigma_k $$ is a $3$-form on $U$ that turns out to be independent of the choice of the orthonormal frame field $X$ on $U$. Hence, it is the restriction to $U$ of a globally defined $3$-form $\Psi$ on $M$ that depends on both the choice of $g$ and the connection $\nabla$.

Are you asking why $\Psi$ is well-defined? That follows from representation theory of $\mathrm{O}(n)$, although one can also define it 'Nomizu-style' as $$ \Psi(Z_1,Z_2,Z_3) = \frac16 \sum_{ijk} \epsilon_{ijk}\,g\bigl(Z_i,\nabla_{Z_j}Z_k), $$ where the sum is over all $1\le i,j,k\le 3$ and $\epsilon_{ijk}$ is the fully anti-symmetric symbol satisfying $\epsilon_{123}=1$.

Are you asking about the properties of $\Psi$ in terms of physical interpretation? I don't know how to answer that. I know that physicists use $\Psi$ as a term in some Lagrangians for the pair $(g,\nabla)$ (and possibly other fields), but I don't know much about that, so I can't help you there.

As for the structure equation above and the notation, a standard reference is Kobayashi and Nomizu. I'd suggest that you look there and see if that answers your questions.