Compatible connections on VBundles

[dario sauro]

Hi,

I have defined a torsionless manifold and I have attached to it a vector VBundle, on which I have defined a metric. I have also defined a covariant derivative on the VBundle which is extended from the one on the base manifold.

Unfortunately, I cannot find a way to define this latter covariant derivative as being compatible with the metric on the VBundle. Are there some subtleties that I should take into account for defining such a metric-compatible covariant derivative on a VBundle?

I sincerely thank in advance everyone who will help me out.

Dario

[Jose]

Hi,

I'm not sure I understand your situation, but let me guess. It always helps when questions have code, to see what you have tried to do.

<< xAct`xTensor`

Define the manifold M with a metric g and its Levi-Civita connection cd. I guess this what you call "a torsionless manifold".

DefManifold[M, 4, {a, b, c, d, f}]

DefMetric[-1, g[-a, -b], cd]

Then define a VBundle, say of dimension 7:

DefVBundle[VB, M, 7, {A, B, C, D, F}]

Define a connection CD on VB, extended from cd, and specifying that it is compatible with the metric g:

DefCovD[CD[-a], VB, ExtendedFrom -> cd, FromMetric -> g]

Then both of these give zero:

In[6]:= cd[-a][g[-b, -c]]
Out[6]= 0

In[7]:= CD[-a][g[-b, -c]]
Out[7]= 0

Is this what you were looking for?

Cheers,

Jose.

[dario sauro]

Hi,

I'm sorry for not having written a piece of code right from the beginning. My problem is associated with defining a covariant derivative which is compatible with the metric tensor ("Gv[a,a1]" below) defined on the vector bundle, as I'm writing below.

I use the following definitions:

DefManifold[M, dimM, {mu,nu,rho,sigma}]

DefMetric[+1, g[-mu,-nu], nabla, {";", "D"},
  CurvatureRelations -> True, Torsion -> False]

DefConstantSymbol[dimfields]

(* I define the VBundle together with its metric Gv *)

DefVBundleWithMetric[fields, M, dimfields, {a,a1,a2,a3,a4,a5}, Gv]

(* I define the covariant derivative on the VBundle such that it is extended from the CovD on the base manifold *)

DefCovD[fieldscovd[-mu], fields, {";", "Der"},
 ExtendedFrom -> nabla, FromMetric -> Gv, CurvatureRelations -> True]

(* I change the options of CollectTensors otherwise I cannot manipulate the most general expressions containing covariant derivatives of tensors valued in the VBundle "fields" *)

SetOptions[CollectTensors, CollectMethod -> ContractMetric]

(* I declare "Gv" to have vanishing covariant derivative  *)

MakeRule[{fieldscovd[-mu]@Gv[a, a1], 0}]
AutomaticRules[Gv, %]

My question is: is it possible to define the covariant derivative on the VBundleWithMetric such that the covariant derivative of the metric defined on the VBundle is always zero? I have circumvented this issue by changing the the options of the command CollectTensors, but in this way I obtain many terms that only differ by the name of the dummy index. I stress the fact that without changing the options of CollectTensors I wasn't able to manipulate expressions involving the covariant derivatives of tensors.

I sincerely thank in advance those who will help me out.

Cheers,

Dario