A manifold admits a huge plurality of connections. If I have a metric (and a few other reasonable preferences) I get a unique metric connection. But if I start with an arbitrary connection, can I find a metric that it could have come from? Why or why not? What implications might this have for differential geometry?

Edit: the visual that I'd had in my head up to now, to make sense of manifolds having many many connections, was of embedding the manifold into Rn and then "squeezing" it: each connection would correspond somehow to a different embedding. It was later I realized (I think) that this implicitly assumes every connection is a metric connection. But, does it really assume that? Is there any sense in which the foregoing is true? (Clearly we are talking about embeddable manifolds, so... second-countable and Hausdorf, I believe).

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    $\begingroup$ In general, no. At the least, the metric need to be torsion free but even that is not enough. See answers of this MO questions. $\endgroup$ Jun 6 '18 at 18:13
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    $\begingroup$ For me, the most concrete example of a torsion connection is to take the left-invariant vector fields $X_i$ on a Lie group to satisfy $\nabla_{X_i}X_j = 0$ for all $i,j$. (This will be a connection with zero curvature.) Then all the Lie group structure constants come in the torsion. $\endgroup$ Jun 6 '18 at 18:52
  • $\begingroup$ There's also this paper: arxiv.org/abs/0804.2698 $\endgroup$
    – exchange
    Sep 2 at 20:28

Short answer, no.

Connections on manifolds can be induced by a metric tensor. These are the usual Christoffel symbols.

There is a condition that is placed on the metric tensor, namely that it be constant w/r to the covariant derivative. This constraint is sometimes used to derive the connection.

One has the freedom to include an anti-symmetric part to the connection called a torsion tensor. While the connection will not transform as a tensor, torsion will. Torsion is a proper tensor.

Torsion does not ruin the covariant constant condition imposed on the metric tensor.

Lastly one can simply allow the above constraint to be violated, leading to what is called non-metricity in the connection. This is the most general type of connection that I've encountered. This can be split into three parts, the metric induced connection, torsion, and whatever is left over.

As an interesting side note Einstein, in his search for a unified field theory included an anti-symmetric part to the metric tensor which leads to a type of torsion field (not the most general type). In GR this is referred to as Moffat torsion named after a physicist who worked with this type of metric model.

In light of this I believe that the comment is not true. One can have a perfectly reasonable metric space and connection with torsion.


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