1

Context

Let $\pi:\text{Fr}(TM)\to M$ be the frame bundle of a smooth manifold $M$.

For simplicity $P:=\text{Fr}(TM)$. I interpret an element $\phi\in P_x$ as linear isomorphism $\mathbb{R}^n\to T_xM$.

On this $GL(n,\mathbb{R})$-principal bundle there is a special form $\theta\in \Omega^1(P,\mathbb{R})$ $$\theta(v):=\phi^{-1}(d\pi(v)),\ \ \ v\in T_\phi P.$$ The form $\theta$ is often called the solder form (or the tautological form). This form makes possible to the define the torsion of a connection $\omega\in \Omega^1(P,\mathfrak{gl}(n,\mathbb{R}))$ of the principal bundle $P$: $$T^\omega(u,v):=d\theta(u^h,v^h)$$ where $u^h$ and $v^h$ are the horizontal components of $u$ and $v$ (with respect to the fixed connection $\omega$).

My problem

I'm trying the prove the torsion structure equation i.e. $$T^\omega(u,v)=d\theta(u,v)+\frac12\left(\omega(u)\cdot\theta(v)-\omega(v)\cdot\theta(u)\right)$$

(the dot $\cdot$ is simply the product of a matrix by a vector).

This should be fairly simple, but I can't do it. I tried explicitating the definitions but it gets messy really fast.

Kandinskij
  • 3,407
  • 2
    Do you know the third bullet point here? The proof is by mimicking the proof of Cartan’s structure equation: consider three cases, (1) is where we evaluate on no vertical vectors (i.e all horizontal), (2) is where we evaluate on atleast one vertical vector, (3) is where we evaluate on atleast two vertical vectors. Now specialize to the case of the frame bundle $P=\text{Fr}(TM)$, and $\rho:\text{GL}(\Bbb{R}^n)\times\Bbb{R}^n\to\Bbb{R}^n$ being evaluation so $\rho’:\text{End}(\Bbb{R}^n)\times\Bbb{R}^n\to\Bbb{R}^n$ is also the evaluation map. – peek-a-boo Aug 15 '24 at 19:49
  • 2
    Also, I’m a little concerned by the factor of $\frac{1}{2}$; I don’t think it should be there (atleast if your definition of the exterior derivative follows the same combinatorial conventions as in Spivak’s Calculus on Manifolds for example). – peek-a-boo Aug 15 '24 at 19:51
  • 1
    oof shouldn’t have bountied this. The above result is pretty standard; I first encountered it as a guided exercise in Dieudonne’s Treatise on Analysis Vol IV, chapter 20.3 problem 1. But since your question, I looked up more friendly sources, and I found it in Loring Tu’s Differential Geometry, Connections, Curvature, and Characteristic Classes, Theorem 31.19. Also, this result appears as Theorem 3.1.5 in David Bleeker’s Gauge Theory and Variational Principles. I’m sure Kobayashi-Nomizu also have this result (and IIRC, their wedge conventions are different, so should match your $1/2$). – peek-a-boo Aug 17 '24 at 19:49
  • 1
    Also, Dieudonne proves (same book and chapter) exactly the torsion structure equation in paragraph 20.6.3 (he does this explicitly and separately because he doesn’t assume knowledge of previously given exercises in the middle of the text). So, I could write up a proof of the result linked above, but that seems like a waste of time. The more important question is whether you know how to apply that result to prove the torsion structure equation? – peek-a-boo Aug 17 '24 at 19:50
  • Thank you for the souce by Loring Tu. If you post it as an answer, I'll give you the bounty. – Kandinskij Aug 18 '24 at 13:08

1 Answers1

1

Converting the comments into an answer: we have the following general theorem

Let $(P,\pi,M,G)$ be a principal bundle with a principal connection with connection 1-form $\omega$, and let $\rho:G\to\text{GL}(V)$ be a linear representation of $G$ on a real finite-dimensional Banach space $V$, and let $\rho’:\mathfrak{g}\times V\to V$ be the bilinear map as induced by the pushforward at the identity $\rho_*:\mathfrak{g}\to\text{End}(V)$. Then, for every $V$-valued $k$-form $\alpha$ on $P$ which is horizontal and $G$-equivariant, we can compute the exterior covariant differential by the formula \begin{align} D\alpha&=d\alpha+\omega\wedge_{\rho’}\alpha, \end{align} where $\wedge_{\rho’}$ is wedge product relative to the bilinear pairing $\rho’$.

Here, $\wedge_{\rho’}$ is the wedge product relative to the bilinear pairing $\rho’$ (see here or here for a much more detailed explanation). Also, the equivariance condition is for $\alpha$ as a map $\bigwedge^k(TP)\to V$, relative to action of $G$ on $\bigwedge^k(TP)$ induced by given the principal action of $G$ on $P$, and the given linear action (via $\rho$) of $G$ on $V$.

Here are the relevant references:

  • Dieudonne, Treatise on Analysis, Vol IV, Chapter 20.3, Problem 1.
  • Loring Tu Differential Geometry: Connections, Curvature, Characteristic Classes, Theorem 31.19
  • David Bleeker Gauge Theory and Variational Principles, Theorem 3.1.5

The torsion structure equation follows immediately since the soldering form is horizontal and equivariant. Alternatively, the proof of the structure equation itself can be found in

  • Dieudonne, Treatise on Analysis, Vol IV, paragraph (20.6.3)
  • Kobayashi, Nomizu Foundations of Differential Geometry, Vol I, page 120 (Theorem 2.4). Note that KN have the factor of $\frac{1}{2}$ because their conventions for the wedge product (hence the exterior derivative) is different.
peek-a-boo
  • 65,833