- #1

- 970

- 3

[tex]V=V^ie_i [/tex]

[tex]dV=dV^ie_i+V^ide_i [/tex]

[tex]dV=\partial_jV^ie_idx^j+V^i \Gamma^{j}_{ir}e_j dx^r [/tex]

which after relabeling indices:

[tex]dV=(\partial_jV^i+V^k \Gamma^{i}_{kj})e_i dx^j [/tex]

so that the covariant derivative is defined as:

[tex]\nabla_j V^i=\partial_jV^i+V^k \Gamma^{i}_{kj}[/tex]

However, the connection coefficient [tex]\Gamma^{i}_{kj} [/tex] is torsion-free by definition, as [tex]de_i=\Gamma^{j}_{ir}e_j dx^r [/tex] implies that

(1) [tex]\partial_r e_i=\Gamma^{j}_{ir}e_j [/tex].

If [tex]e_i=\partial_i[/tex] then since [tex]\partial_i\partial_r=\partial_r\partial_i[/tex] then by (1):

[tex]\Gamma^{j}_{ir}e_j=\Gamma^{j}_{ri}e_j [/tex]

or that the bottom two indices are symmetric which is the torsion-free condition.

I have two questions. Is the equation [tex]\partial_r e_i=\Gamma^{j}_{ir}e_j [/tex] true in general for any connection? And also, where did the torsion-free assumption enter into the derivation above?