Cartan structural equations

To deduce the Cartan structural equations in a coordinated frame we are going to use the definition of the Christoffel symbolsMathworldPlanetmathPlanetmath (connection coefficients) and where we always are going to use the Einstein sum convention:


and the curvature tensor


where X,Y,Z are any three vector fields in a riemannian manifoldMathworldPlanetmath with the Levi-Civita connectionMathworldPlanetmath .

First, we define through the relation Xi=ωsi(X)s a set of scalar function ωsi which are easily to see that they actually are 1-forms. We observe that ωsi(j)=Γsij.

They satisfy skew-symmetry rule: ωsi=-ωis, which arises from the covariant constancy of the metric tensor gkl i.e.

0 = Xgkl
= Xk,l
= Xk,l+k,Xl
= ωsk(X)s,l+k,ωsl(X)s
= ωsk(X)gsl+ωsl(X)gks
0 = ωlk(X)+ωkl(X)

that last equation is valid for each vector field X, then ωlk=-ωkl.

Next we define through the relation


the scalars Ωsi(X,Y) which are the so called connection 2-forms. That they are really 2-forms is an easy caligraphic exercise.

Now by the use of the Riemann curvature tensorMathworldPlanetmath above we see

R(X,Y)i = XYi-YXi-[X,Y]i
= X(ωsi(Y)s)-Y(ωsi(X)s)-ωsi[X,Y]s
= X(ωsi(Y))s+ωsi(Y)Xs-Y(ωsi(X)s-ωsi(X)Ys-ωsi[X,Y]s
= X(ωsi(Y))s+ωsi(Y)ωts(X)t-Y(ωsi(X)s-ωsi(X)ωts(Y)t-ωsi[X,Y]s
= [X(ωsi(Y))+ωti(Y)ωst(X)-Y(ωsi(X))-ωti(X)ωst(Y)-ωsi[X,Y]]s
Ωsi(X,Y)s = [X(ωsi(Y))-Y(ωsi(X))-ωsi[X,Y]+ωst(X)ωti(Y)-ωst(Y)ωti(X)]s

In this last relation we recognize -in the first three terms- the exterior derivativeMathworldPlanetmath of ωsi evaluated at (X,Y) i.e.


and in the last two terms its wedge product


all these for any two fields X,Y. Hence


which is called the second Cartan structural equation for the coordinated frame field i.

More interesting things happen in an an-holonomic basis.

Title Cartan structural equations
Canonical name CartanStructuralEquations
Date of creation 2013-03-22 17:35:46
Last modified on 2013-03-22 17:35:46
Owner juanman (12619)
Last modified by juanman (12619)
Numerical id 12
Author juanman (12619)
Entry type Result
Classification msc 53A45
Classification msc 58A12
Classification msc 58A10