Not signed in (Sign In)

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

  • Sign in using OpenID

Site Tag Cloud

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

Welcome to nForum
If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).
    • CommentRowNumber1.
    • CommentAuthorBruce Bartlett
    • CommentTimeSep 22nd 2013

    I’d like to add some sentences describing a geometric understanding of the “torsion” of a connection on a Riemannian manifold to torsion. But perhaps my understanding of torsion is wrong, so I’m running it by you guys first. I wrote it down on math overflow and I’m curious what people think.

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 22nd 2013

    There’s probably quite a lot of material worth borrowing from that thread. What about organising things as Chris Schommer-Pries does at a more general level than Riemannian geometry?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2013
    • (edited Sep 22nd 2013)

    By the way, what Chris S.-P. indicates in that entry on the integrability of G-structures is alluding to what is explained in section 2.6 of

    • Dominic Joyce, Compact manifolds with special holonomy, Oxford 2000 .

    This perspective reminds me of what I was recently after on MO here,

    But as far as I can see, for Bruce’s purposes the perspective of integrability of G-structures is going a bit off on a tangent. I think specifically for torsion of metric connections in the context of gravity, there is a clear best perspective, and its the one indicated on the nLab at torsion of a metric connection:

    namely a metric connection is precisely a Cartan connection for the inclusion of the orthogobal/Lorentz group into the Euclidean/Poincaré group. Such a connection has precisely two different curvature components, corresponding to the two pieces in the extension

    O(d)Iso( d) d. O(d) \longrightarrow Iso(\mathbb{R}^d) \longrightarrow \mathbb{R}^d \,.

    The two curvature components are the Riemann curvature and the torsion.

    And this is pretty much what Bruce is after, I think.

    • CommentRowNumber4.
    • CommentAuthorBruce Bartlett
    • CommentTimeSep 22nd 2013

    David: I agree there’s lots of stuff going on there. In the long term I’m going to try understand CSP’s comment. Right now I’m after a more down-to-earth geometric interpretation. Urs: That’s interesting I will have to think about that.

    I think there are some problems with my interpretation of torsion. For instance, it would imply that a metric connection on a surface cannot have torsion (because once you fix a vector in a frame e 1,e 2e_1, e_2, you cannot rotate the remaining vector). I may have been making the error that it is only in odd dimensions that a rotation has an “axis of rotation”.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2013
    • (edited Sep 22nd 2013)

    I’m going to try understand CSP’s comment

    It’s Dominic Joyces’s observation, explained in section 2.6 of his book “Compact manifolds with special holonomy”

    Urs: That’s interesting I will have to think about that.

    You must have thought of that already, as this is Cartan’s definition of torsion.

    I may have been making the error that it is only in odd dimensions that a rotation has an “axis of rotation”.

    No, an axis of rotation only exists precisely in 3 dimensions. In all other dimensions the axis is replaced by the corresponding Poincaré dual of the given surface of rotation.

    • CommentRowNumber6.
    • CommentAuthorBruce Bartlett
    • CommentTimeSep 23rd 2013

    Yes, apologies for getting rotations wrong. Regarding Cartan’s definition of torsion - I have to admit that I haven’t thought much about the generalized case of Cartan connections, I mainly only know about connections on a principal bundle which is a special case I know. I don’t berate myself for this, since Cartan connections aren’t dealt with in the standard books on differential geometry (eg. Spivak). But you’re right, I’m sure they give a nice explanation of torsion. I will just need some time to get my head around that.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeSep 24th 2013

    A Cartan connection is just a principal connection subject to the constraint that at each point it identifies the tangent space with the quotient Lie algebra.

    Textbooks may not have it, but the nLab does: Cartan connection. ;-)

    It’s super-simple. But it’s also true that there are many discussions of Cartan connections out there which make it look more intricate.

    In particular, physicists have been using Cartan connections for ages, without using the term. This is just what the first order formulation of gravity is: an Iso(d1,1)Iso(d-1,1)-connection such that it identifies at each point the tangent space of spacetime with an d\mathbb{R}^d. That’s all.

    And torsion is just the part of the curvature 2-form with values in the d\mathbb{R}^d-factor. This statement is manifestly Cartan’s (and hence the standard) definition of torsion.