Investigating the Value of [itex]\omega_{c}^{ab}[itex]

[m1rohit]

Homework Statement

I want to know wether the component \omega_{c}^{ab}of the spin connection is zero or not?<br /> <br /> <h2>Homework Equations</h2><br /> de^{a}=-\omega^{a}_{b}\wedge e^{b}<br /> <br /> <br /> <h2>The Attempt at a Solution</h2><br /> de^{a}=-\omega^{a}_{b}\wedge e^{b}<br /> $=-\omega^{a}_{b,\nu} e^{b}_{\mu} dx^{\nu}\wedge dx^{\mu}$<br /> $=-\omega^{a}_{b,c} e^{b}_{[\mu}e^{c}_{\nu]} dx^{\nu}\wedge dx^{\mu}$<br /> as it is antisymmetric in $\mu$ and $\nu$.So it is also antisymmetric in b and c.Thus one can conclude from here.

 

[nrqed]

Homework Statement

I want to know wether the component \omega_{c}^{ab} of the spin connection is zero or not?

Homework Equations

de^{a}=-\omega^{a}_{b}\wedge e^{b}

The Attempt at a Solution

de^{a}=-\omega^{a}_{b}\wedge e^{b}
$=-\omega^{a}_{b,\nu} e^{b}_{\mu} dx^{\nu}\wedge dx^{\mu}$
$=-\omega^{a}_{b,c} e^{b}_{[\mu}e^{c}_{\nu]} dx^{\nu}\wedge dx^{\mu}$
as it is antisymmetric in $\mu$ and $\nu$.So it is also antisymmetric in b and c.Thus one can conclude from here.

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