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[parent] Cartan structural equations (Result)

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

$\displaystyle \nabla_{\partial_ i}\partial_j={\Gamma^s}_{ij}\partial_s$
and the curvature tensor
$\displaystyle R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$
where $ X,Y,Z$ are any three vector fields in a riemannian manifold $ \cal{M}$ with the Levi-Civita connection $ \nabla$.

First, we define through the relation $ \nabla_X\partial_i={\omega^s}_i(X)\partial_s$ a set of scalar function $ {\omega^s}_i$ which are easily to see that they actually are 1-forms. We observe that $ {\omega^s}_i(\partial_j)={\Gamma^s}_{ij}$.

They satisfy skew-symmetry rule: $ \omega_{si}=-\omega_{is}$, which arises from the covariant constancy of the metric tensor $ g_{kl}$ i.e.

0 $\displaystyle =$ $\displaystyle \nabla_{X}g_{kl}$  
  $\displaystyle =$ $\displaystyle \nabla_{X}\langle\partial_k,\partial_l\rangle$  
  $\displaystyle =$ $\displaystyle \langle\nabla_{X}\partial_k,\partial_l\rangle+\langle\partial_k,\nabla_{X}\partial_l\rangle$  
  $\displaystyle =$ $\displaystyle \langle{\omega^s}_k(X)\partial_s,\partial_l\rangle+\langle\partial_k,{\omega^s}_l(X)\partial_s\rangle$  
  $\displaystyle =$ $\displaystyle {\omega^s}_k(X)g_{sl}+{\omega^s}_l(X)g_{ks}$  
0 $\displaystyle =$ $\displaystyle \omega_{lk}(X)+\omega_{kl}(X)$  

that last equation is valid for each vector field $ X$, then $ \omega_{lk}=-\omega_{kl}$.

Next we define through the relation

$\displaystyle R(X,Y)\partial_i={\Omega^s}_i(X,Y)\partial_s$
the scalars $ {\Omega^s}_i(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 tensor above we see

$\displaystyle R(X,Y)\partial_i$ $\displaystyle =$ $\displaystyle \nabla_X\nabla_Y\partial_i-\nabla_Y\nabla_X\partial_i-\nabla_{[X,Y]}\partial_i$  
  $\displaystyle =$ $\displaystyle \nabla_X({\omega^s}_i(Y)\partial_s)-\nabla_Y({\omega^s}_i(X)\partial_s)-{\omega^s}_i[X,Y]\partial_s$  
  $\displaystyle =$ $\displaystyle X({\omega^s}_i(Y))\partial_s+{\omega^s}_i(Y)\nabla_X\partial_s- Y... ...s}_i(X)\partial_s-{\omega^s}_i(X)\nabla_Y\partial_s-{\omega^s}_i[X,Y]\partial_s$  
  $\displaystyle =$ $\displaystyle X({\omega^s}_i(Y))\partial_s+{\omega^s}_i(Y){\omega^t}_s(X)\parti... ...\partial_s-{\omega^s}_i(X){\omega^t}_s(Y)\partial_t-{\omega^s}_i[X,Y]\partial_s$  
  $\displaystyle =$ $\displaystyle [X({\omega^s}_i(Y))+{\omega^t}_i(Y){\omega^s}_t(X)-Y({\omega^s}_i(X))-{\omega^t}_i(X){\omega^s}_t(Y)-{\omega^s}_i[X,Y]]\partial_s$  
$\displaystyle {\Omega^s}_i(X,Y)\partial_s$ $\displaystyle =$ $\displaystyle [X({\omega^s}_i(Y))-Y({\omega^s}_i(X))-{\omega^s}_i[X,Y]+{\omega^s}_t(X){\omega^t}_i(Y)-{\omega^s}_t(Y){\omega^t}_i(X)]\partial_s$  

In this last relation we recognize -in the first three terms- the exterior derivative of $ {\omega^s}_i$ evaluated at $ (X,Y)$ i.e.
$\displaystyle d{\omega^s}_i(X,Y)=X({\omega^s}_i(Y))-Y({\omega^s}_i(X))-{\omega^s}_i[X,Y]$
and in the last two terms its wedge product
$\displaystyle {\omega^s}_t\wedge{\omega^t}_i(X,Y)={\omega^s}_t(X){\omega^t}_i(Y)-{\omega^s}_t(Y){\omega^t}_i(X)$
all these for any two fields $ X,Y$. Hence
$\displaystyle {\Omega^s}_i=d{\omega^s}_i+{\omega^s}_t\wedge{\omega^t}_i$
which is called the second Cartan structural equation for the coordinated frame field $ \partial_i$.

More interesting things happen in an an-holonomic basis.



"Cartan structural equations" is owned by juanman.
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Keywords:  Cartan Calculus, differential forms, spin connection, holonomy, an-holonomy

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Cross-references: basis, fields, wedge product, terms, exterior derivative, Riemann curvature tensor, connection, equation, metric tensor, 1-forms, function, scalar, relation, Levi-Civita connection, Riemannian manifold, vector fields, tensor, curvature, sum, Christoffel symbols, frame
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This is version 9 of Cartan structural equations, born on 2007-10-22, modified 2007-10-23.
Object id is 10010, canonical name is CartanEstructuralEquations.
Accessed 656 times total.

Classification:
AMS MSC: 58A10 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Differential forms)
 58A12 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: de Rham theory)
 53A45 (Differential geometry :: Classical differential geometry :: Vector and tensor analysis)
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