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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?

Thanks

flag	asked Nov 26 2010 at 8:36 Jerzy Kowalski-Glikman 21●1

“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 2010 at 9:05

Sure, but I can imbed it in $R^5$ and use the induced metric. – Jerzy Kowalski-Glikman Nov 26 2010 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 2010 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 2010 at 16:57

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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 2010 at 17:25

Richard Montgomery · Answer 1 · 2010-11-27 00:55:01Z UTC

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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.

answered Nov 27 2010 at 0:55

Richard Montgomery
3,193●8●18

connections on AN(3) Lie group

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