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

[m1rohit]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

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

 

[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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