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I have run across the existence of torsion as I study Reimannian geometry. I also know that in the case of Reimannian geometry, we can always find a unique metric-preserving connection with zero torsion: the Levi-Cevita connection.

This begs the question, why is torsion a fruitful concept? I have found certain answers on math.se which provide examples of connections with torsion that look highly unnatural, like this one about torsion in two dimensions. What do we as mathematicians gain by studying torsion? Is there a single "natural" example of torsion?

This Math overflow question: what is torsion intuitively seems to have fantastic answers that I cannot access - I simply do not know enough math, in particular, Lie groups and solder forms. Is there some way to "elaborate" the answers there with an example in 2D or 3D such that the essence is retained?

I have often seen this picture:

torsion-picture

While this picture shows us what torsion is after defining it, it doesn't tell us why we would care to pick such a connection in the first place! So this is not a satisfactory answer for me right now. I want to understand why we even want torsion.

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    $\begingroup$ I suspect this MO question and its answers may be relevant. (Granted, it's about understanding rather than applying torsion, but I still think it may be of interest - especially re: your last sentence.) $\endgroup$ – Noah Schweber Sep 20 '19 at 16:35
  • $\begingroup$ I found that right after I posted this :) Unfortunately, most of the answers there are far too "high brow" for me: I don't yet know lie groups, or an intuition for exact sequences :( Is there some way to "elaborate" the answers there with an example in 2D or 3D such that the essence is retained? $\endgroup$ – Siddharth Bhat Sep 20 '19 at 16:38
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    $\begingroup$ This isn't anywhere near my competency region (neighborhood? cul-de-sac?); I just remembered it being an awesome-if-impenetrable question I once saw. Hopefully someone else will chime in. $\endgroup$ – Noah Schweber Sep 20 '19 at 16:39
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    $\begingroup$ The question is really why connection is interesting. There are connections besides the Levi-Civita connection. In some situation those connections are more natural to work with. The Chern connection is one such example, when one has a harmitian metric on a complex manifold. $\endgroup$ – Arctic Char Sep 20 '19 at 20:12
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Just as in Riemannian geometry, where it is (thoroughly) helpful to have a connection that preserves the metric $g$, for any structure on the tangent bundle it is convenient for some purposes to consider connections that preserve that structure.

Example A contact distribution is a maximally nonintegrable hyperplane distribution $\bf H$ on an odd-dimensional manifold $M$ (say, $\dim M = 2 m + 1 \geq 3$). Here, maximally nonintegrable means that at any point $p \in M$, $$\{[X, Y]_p : X, Y \in \Gamma({\bf H})\} = T_p M,$$ or equivalently, that for any (every) form $\theta \in \Gamma(T^*M)$ such that $\ker \theta = {\bf H}$ (this form is unique up to scale), $$\theta \wedge d\theta \wedge \cdots \wedge d\theta$$ vanishes nowhere. (Such a form is called a contact form.)

So, let's look at connections that preserve ${\bf H}$---this is useful (essential) in studying the geometry of contact distributions equipped with additional structure (the best-studied example is CR geometry). In terms of the contact form $\theta$, preserving ${\bf H}$ means precisely that $$\nabla \theta = \eta \otimes \theta$$ for some $1$-form $\eta \in \Gamma(TM)$ (that depends on $\theta$). But if $\nabla$ were torsion-free, a standard identity (see, e.g., Kobayashi & Nomizu, Foundations of Differential Geometry, Volume 1, Corollary 8.6) yields $$d\theta = 2 \operatorname{Alt}(\nabla \theta) = 2 \operatorname{Alt}(\eta \otimes \theta) = \eta \wedge \theta .$$ But then $$\theta \wedge d\theta = \theta \wedge (\eta \wedge \theta) = 0 ,$$ which contradicts that $\theta$ is a contact form. (In fact, appealing to the differential ideal characterization of integrable distributions shows that this argument demonstrates an even stronger fact, namely that if a torsion-free connection preserves a hyperplane distribution, that distribution must be integrable.)

There are many other examples. To expand briefly on a comment of Arctic Char: If $(g, J)$ is an almost Hermitian structure, then just like we can single out a preferred connection on a Riemannian manifold (the Levi-Civita connection), we can also single out a preferred connection associated to $(g, J)$---sometimes called the Chern connection---by imposing:

  • $\nabla g = 0$ (that $\nabla$ is a metric connection for $g$)
  • $\nabla J = 0$
  • $(T^\nabla)^{1, 1} = 0$ (this condition is that a particular invariant part of the torsion tensor $T^\nabla$ vanishes).

But if $J$ is not integrable, this preferred connection is not torsion-free.

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  • $\begingroup$ Thank you for this answer! Would it be possible to list out other notions on a manifold one might want to "preserve" -- like the metric, or contact distribution? This way, I feel I could get a birds eye view of what structures are interesting, and how torsion helps. $\endgroup$ – Siddharth Bhat Sep 21 '19 at 11:31
  • $\begingroup$ I'm not sure an exhaustive list is reasonable, but the examples in my answer and their variations are probably the most important examples. $\endgroup$ – Travis Willse Sep 21 '19 at 18:13
  • $\begingroup$ For example, the argument that a connection preserving a contact distribution must have nonzero torsion applies mutatis mutandis to any nonintegrable tangent distribution $\bf D$: If we locally write $\bf D=\ker\{\theta^1,\ldots,\theta^k\}$ for pointwise independent forms $\theta^i$, then $\nabla$ preserves $\bf D$ iff there are forms $\lambda^j{}_i$ such that $\nabla\theta_i=\sum_j\lambda^j{}_i\otimes\theta_j$. But if $\nabla$ torsion-free, skewing gives $d\theta_i=\sum_j\lambda^j{}_i\wedge\theta_j$, and again $\ker\{\theta_i\}={\bf D}$ is integrable, a contradiction. $\endgroup$ – Travis Willse Sep 21 '19 at 18:17

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