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I am a physicist, and thus my question might be not precise and/or even not well posed. The problem I encounter is to characterize all `natural' connections on the group manifold AN(3). AN(3) is a 4-dimensional Lie group with the Lie algebra generated by $x_i$, $i=1,2,3$, $t$ satisfying

$[t, x_i]=x_i$, $[x_i, x_j]=0$

The AN(3) group is a 4-dimensional manifold, so I can clearly construct a Levi-Civita connection on it (with non-zero curvature and zero torsion). I can also take Maurer-Cartan forms (zero curvature and non-zero torsion). Are there any other natural ones?

More generally. As a manifold the AN(3) group is a submanifold of de Sitter space $SO(4,1))/SO(3,1)$. The Levi-Civita connection on de Sitter is well known in physics in the context of cosmology, for example, but are there any other natural/interesting ones?


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“The AN(3) group is a 4-dimensional manifold, so I can clearly construct a Levi-Civita connection on it (with non-zero curvature and zero torsion).” Well, you need a riemannian metric on it in order to have a Levi-Civita connection. – Guillaume Brunerie Nov 26 '10 at 9:05
Sure, but I can imbed it in $R^5$ and use the induced metric. – Jerzy Kowalski-Glikman Nov 26 '10 at 11:18
I don't know what AN(3) is; do you have a more concrete description of the group itself and not just its Lie algebra? How do you imbed it in $R^5$? Finally, your description of how you construct the Levi-Civita connection is rather vague and does not sound natural to me. Could you provide more detail on that? – Deane Yang Nov 26 '10 at 15:48
You can find the explicit representation of the algebra and group in our paper arXiv:0706.3658 [hep-th] (section IIA). From there you can also easily reconstruct the metric and Levi-Civita connection. The metric turns out to be $-dt^2 +e^t\, \sum (dx^i)^2$ – Jerzy Kowalski-Glikman Nov 26 '10 at 16:57
I guess the question is what you mean by 'natural'. On homogeneous spaces (of which Lie groups are of course examples) there is a notion of invariant connection, as described in, say, Kobayashi-Nomizu. Another possible notion of 'natural' connection would be the Levi-Civita connection of some left-invariant metric. – José Figueroa-O'Farrill Nov 26 '10 at 17:25

The space of connections forms an affine space, so you can take convex combinations and average. Thus you can take the average of your connections also.

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