Back
Thanks to Bonbien Vargas and Nail Ussembayev for helping me with this section.
Riemann curvature of a metric-cancelling, non-coordinate-basis, (torsion-free?) connection of a vector:
$R(x,y)z$
$= (\left[ \nabla_x, \nabla_y \right] - \nabla_{[x,y]}) z$
$= \nabla_x \nabla_y z - \nabla_y \nabla_x z - \nabla_{[x,y]} z$
$= x^a \nabla_a ( y^b \nabla_b ( z^d e_d ) )
- y^b \nabla_b ( x^a \nabla_a ( z^d e_d ) )
- (x^a e_a(y^b) - y^a e_a(x^b) + x^a y^c {c_{ac}}^b) \nabla_b (z^d e_d)$
$= x^a \nabla_a ( y^b (\nabla_b z^d e_d + z^d \nabla_b e_d ) )
- y^b \nabla_b ( x^a (\nabla_a z^d e_d + z^d \nabla_a e_d ) )
- (x^a e_a(y^b) - y^a e_a(x^b) + x^a y^c {c_{ac}}^b) (\nabla_b z^d e_d + z^d \nabla_b e_d)$
$= x^a \nabla_a ( y^b (e_b(z^d) + z^c {\Gamma^d}_{bc} ) e_d )
- y^b \nabla_b ( x^a (e_a(z^d) + z^c {\Gamma^d}_{ac} ) e_d )
- (x^a e_a(y^b) - y^a e_a(x^b) + x^a y^u {c_{au}}^b) (e_b(z^c) + z^d {\Gamma^c}_{bd}) e_c$
$= x^a e_a ( y^b (e_b(z^d) + z^c {\Gamma^d}_{bc} ) ) e_d
+ x^a y^b (e_b(z^d) + z^c {\Gamma^d}_{bc} ) {\Gamma^u}_{ad} e_u
- y^b e_b ( x^a (e_a(z^d) + z^c {\Gamma^d}_{ac} ) ) e_d
- y^b x^a (e_a(z^d) + z^c {\Gamma^d}_{ac} ) {\Gamma^u}_{bd} e_u
- x^a e_a(y^b) (e_b(z^c) + z^d {\Gamma^c}_{bd}) e_c
+ y^a e_a(x^b) (e_b(z^c) + z^d {\Gamma^c}_{bd}) e_c
- x^a y^u {c_{au}}^b (e_b(z^c) + z^d {\Gamma^c}_{bd}) e_c$
$= x^a y^b z^d (
e_a({\Gamma^c}_{bd})
- e_b({\Gamma^c}_{ad})
+ {\Gamma^c}_{au} {\Gamma^u}_{bd}
- {\Gamma^c}_{bu} {\Gamma^u}_{ad}
- {\Gamma^c}_{ud} {c_{ab}}^u
) e_c$
$= x^a y^b z^d (
2 e_{[a} ({\Gamma^c}_{b]d})
+ 2 {\Gamma^c}_{[a|u} {\Gamma^u}_{b]d}
- {\Gamma^c}_{ud} {c_{ab}}^u
) e_c$
$= x^a y^b z^d {R^c}_{dab} e_c$
For the Riemann curvature tensor index terms:
${R^c}_{dab} =
2 e_{[a} ({\Gamma^c}_{b]d})
+ 2 {\Gamma^c}_{[a|u} {\Gamma^u}_{b]d}
- {\Gamma^c}_{ud} {c_{ab}}^u
$
${R^c}_{dab} =
e_a ({\Gamma^c}_{bd})
- e_b ({\Gamma^c}_{ad})
+ {\Gamma^c}_{au} {\Gamma^u}_{bd}
- {\Gamma^c}_{bu} {\Gamma^u}_{ad}
- {\Gamma^c}_{ud} {c_{ab}}^u
$
Easy to see from this that ${R^a}_{bcd} = {R^a}_{b[cd]}$
Riemann curvature of the coordinate-basis Levi-Civita connection of a vector:
${\tilde{R}^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}} =
\partial_\tilde{a} ({\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{d}})
- \partial_\tilde{b} ({\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{d}})
+ {\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{u}} {\tilde{\Gamma}^\tilde{u}}_{\tilde{b}\tilde{d}}
- {\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{u}} {\tilde{\Gamma}^\tilde{u}}_{\tilde{a}\tilde{d}}
$
Transformation of the non-coordinate, metric-cancelling, torsion-free connection to the coordinate basis:
${R^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}} =
{e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} (
e_a({\Gamma^c}_{bd})
- e_b({\Gamma^c}_{ad})
+ {\Gamma^{c}}_{{a}{u}} {\Gamma^{u}}_{{b}{d}}
- {\Gamma^{c}}_{{b}{u}} {\Gamma^{u}}_{{a}{d}}
- {\Gamma^{c}}_{{u}{d}} {c_{{a}{b}}}^{u}
)
$
...using ${\Gamma^a}_{bc} = {e^a}_\tilde{a} {e_b}^\tilde{b} {e_c}^\tilde{c} {\tilde{\Gamma}^\tilde{a}}_{\tilde{b}\tilde{c}} + {e^a}_\tilde{a} e_b({e_c}^\tilde{a})$...
${R^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}} =
{e_c}^\tilde{c} {e^b}_\tilde{b} {e^d}_\tilde{d} \partial_\tilde{a} (
{e^c}_\tilde{r} {e_b}^\tilde{q} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{s}} + {e^c}_\tilde{r} e_b({e_d}^\tilde{r})
)
$
$
- {e_c}^\tilde{c} {e^a}_\tilde{a} {e^d}_\tilde{d} \partial_\tilde{b} (
{e^c}_\tilde{r} {e_a}^\tilde{p} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{s}} + {e^c}_\tilde{r} e_a({e_d}^\tilde{r})
)
$
$
+ {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} (
{e^c}_\tilde{r} {e_a}^\tilde{p} {e_u}^\tilde{u} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{u}} + {e^c}_\tilde{r} e_a({e_u}^\tilde{r})
) (
{e^u}_\tilde{v} {e_b}^\tilde{q} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{v}}_{\tilde{q}\tilde{s}} + {e^u}_\tilde{v} e_b({e_d}^\tilde{v})
)
$
$
- {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} (
{e^c}_\tilde{r} {e_b}^\tilde{q} {e_u}^\tilde{u} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{u}} + {e^c}_\tilde{r} e_b({e_u}^\tilde{r})
) (
{e^u}_\tilde{v} {e_a}^\tilde{p} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{v}}_{\tilde{p}\tilde{s}} + {e^u}_\tilde{v} e_a({e_d}^\tilde{v})
)
$
$
- {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} (
{e^c}_\tilde{r} {e_u}^\tilde{u} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{u}\tilde{s}} + {e^c}_\tilde{r} e_u({e_d}^\tilde{r})
) {c_{{a}{b}}}^{u}
$
${R^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}} =
{e_c}^\tilde{c} {e^b}_\tilde{b} {e^d}_\tilde{d} \partial_\tilde{a} \left(
{e^c}_\tilde{r} {e_b}^\tilde{q} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{s}}
\right)
+ {e_c}^\tilde{c} {e^b}_\tilde{b} {e^d}_\tilde{d} \partial_\tilde{a} \left(
{e^c}_\tilde{r} e_b({e_d}^\tilde{r})
\right)
$
$
- {e_c}^\tilde{c} {e^a}_\tilde{a} {e^d}_\tilde{d} \partial_\tilde{b} \left(
{e^c}_\tilde{r} {e_a}^\tilde{p} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{s}}
\right)
- {e_c}^\tilde{c} {e^a}_\tilde{a} {e^d}_\tilde{d} \partial_\tilde{b} \left(
{e^c}_\tilde{r} e_a({e_d}^\tilde{r})
\right)
$
$
+ {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{r} {e_a}^\tilde{p} {e_u}^\tilde{u} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{u}} {e^u}_\tilde{v} {e_b}^\tilde{q} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{v}}_{\tilde{q}\tilde{s}}
+ {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{r} e_a({e_u}^\tilde{r}) {e^u}_\tilde{v} {e_b}^\tilde{q} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{v}}_{\tilde{q}\tilde{s}}
+ {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{r} {e_a}^\tilde{p} {e_u}^\tilde{u} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{u}} {e^u}_\tilde{v} e_b({e_d}^\tilde{v})
+ {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{r} e_a({e_u}^\tilde{r}) {e^u}_\tilde{v} e_b({e_d}^\tilde{v})
$
$
- {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{r} {e_b}^\tilde{q} {e_u}^\tilde{u} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{u}} {e^u}_\tilde{v} {e_a}^\tilde{p} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{v}}_{\tilde{p}\tilde{s}}
- {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{r} e_b({e_u}^\tilde{r}) {e^u}_\tilde{v} {e_a}^\tilde{p} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{v}}_{\tilde{p}\tilde{s}}
- {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{r} {e_b}^\tilde{q} {e_u}^\tilde{u} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{u}} {e^u}_\tilde{v} e_a({e_d}^\tilde{v})
- {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{r} e_b({e_u}^\tilde{r}) {e^u}_\tilde{v} e_a({e_d}^\tilde{v})
$
$
- {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{r} {e_u}^\tilde{u} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{u}\tilde{s}} {c_{{a}{b}}}^{u}
- {e_c}^\tilde{c} {e^d}_\tilde{d} {e^a}_\tilde{a} {e^b}_\tilde{b} {e^c}_\tilde{r} e_u({e_d}^\tilde{r}) {c_{{a}{b}}}^{u}
$
${R^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}} =
{e_c}^\tilde{c} {e^b}_\tilde{b} {e^d}_\tilde{d} (
\partial_\tilde{a} {e^c}_\tilde{r} {e_b}^\tilde{q} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{s}}
+ {e^c}_\tilde{r} \partial_\tilde{a} {e_b}^\tilde{q} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{s}}
+ {e^c}_\tilde{r} {e_b}^\tilde{q} \partial_\tilde{a} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{s}}
+ {e^c}_\tilde{r} {e_b}^\tilde{q} {e_d}^\tilde{s} \partial_\tilde{a} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{s}}
)
+ {e_c}^\tilde{c} {e^b}_\tilde{b} {e^d}_\tilde{d} \partial_\tilde{a} (
{e^c}_\tilde{r} {e_b}^\tilde{q} \partial_\tilde{q}({e_d}^\tilde{r})
)
$
$
- {e_c}^\tilde{c} {e^a}_\tilde{a} {e^d}_\tilde{d} (
\partial_\tilde{b} {e^c}_\tilde{r} {e_a}^\tilde{p} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{s}}
+ {e^c}_\tilde{r} \partial_\tilde{b} {e_a}^\tilde{p} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{s}}
+ {e^c}_\tilde{r} {e_a}^\tilde{p} \partial_\tilde{b} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{s}}
+ {e^c}_\tilde{r} {e_a}^\tilde{p} {e_d}^\tilde{s} \partial_\tilde{b} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{s}}
)
- {e_c}^\tilde{c} {e^a}_\tilde{a} {e^d}_\tilde{d} \partial_\tilde{b} \left(
{e^c}_\tilde{r} {e_a}^\tilde{p} \partial_\tilde{p}({e_d}^\tilde{r})
\right)
$
$
+ {\tilde{\Gamma}^\tilde{c}}_{\tilde{a}u} {\tilde{\Gamma}^u}_{\tilde{b}\tilde{d}}
+ \partial_\tilde{a}({e_u}^\tilde{c}) {\tilde{\Gamma}^u}_{\tilde{b}\tilde{d}}
+ {\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{v}} {e^d}_\tilde{d} \partial_\tilde{b}({e_d}^\tilde{v})
+ \partial_\tilde{a}({e_u}^\tilde{c}) {e^u}_\tilde{v} {e^d}_\tilde{d} \partial_\tilde{b}({e_d}^\tilde{v})
$
$
- {\tilde{\Gamma}^\tilde{c}}_{\tilde{b}u} {\tilde{\Gamma}^u}_{\tilde{a}\tilde{d}}
- \partial_\tilde{b}({e_u}^\tilde{c}) {\tilde{\Gamma}^u}_{\tilde{a}\tilde{d}}
- {\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{v}} {e^d}_\tilde{d} \partial_\tilde{a}({e_d}^\tilde{v})
- \partial_\tilde{b}({e_u}^\tilde{c}) {e^u}_\tilde{v} {e^d}_\tilde{d} \partial_\tilde{a}({e_d}^\tilde{v})
$
$
- {\tilde{\Gamma}^\tilde{c}}_{u\tilde{d}} {c_{\tilde{a}\tilde{b}}}^{u}
- {e^d}_\tilde{d} e_u({e_d}^\tilde{c}) {c_{\tilde{a}\tilde{b}}}^{u}
$
${R^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}} =
{e_c}^\tilde{c} {e^b}_\tilde{b} {e^d}_\tilde{d} (
\partial_\tilde{a} {e^c}_\tilde{r} {e_b}^\tilde{q} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{s}}
+ {e^c}_\tilde{r} \partial_\tilde{a} {e_b}^\tilde{q} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{s}}
+ {e^c}_\tilde{r} {e_b}^\tilde{q} \partial_\tilde{a} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{s}}
+ {e^c}_\tilde{r} {e_b}^\tilde{q} {e_d}^\tilde{s} \partial_\tilde{a} {\tilde{\Gamma}^\tilde{r}}_{\tilde{q}\tilde{s}}
)
$
$
- {e_c}^\tilde{c} {e^a}_\tilde{a} {e^d}_\tilde{d} (
\partial_\tilde{b} {e^c}_\tilde{r} {e_a}^\tilde{p} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{s}}
+ {e^c}_\tilde{r} \partial_\tilde{b} {e_a}^\tilde{p} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{s}}
+ {e^c}_\tilde{r} {e_a}^\tilde{p} \partial_\tilde{b} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{s}}
+ {e^c}_\tilde{r} {e_a}^\tilde{p} {e_d}^\tilde{s} \partial_\tilde{b} {\tilde{\Gamma}^\tilde{r}}_{\tilde{p}\tilde{s}}
)
$
$
+ {e_c}^\tilde{c} {e^b}_\tilde{b} {e^d}_\tilde{d} (
\partial_\tilde{a} {e^c}_\tilde{r} {e_b}^\tilde{q} \partial_\tilde{q}({e_d}^\tilde{r})
+ {e^c}_\tilde{r} \partial_\tilde{a} {e_b}^\tilde{q} \partial_\tilde{q}({e_d}^\tilde{r})
+ {e^c}_\tilde{r} {e_b}^\tilde{q} \partial_\tilde{a} \partial_\tilde{q}({e_d}^\tilde{r})
)
$
$
- {e_c}^\tilde{c} {e^a}_\tilde{a} {e^d}_\tilde{d} (
\partial_\tilde{b} {e^c}_\tilde{r} {e_a}^\tilde{p} \partial_\tilde{p}({e_d}^\tilde{r})
+ {e^c}_\tilde{r} \partial_\tilde{b} {e_a}^\tilde{p} \partial_\tilde{p}({e_d}^\tilde{r})
+ {e^c}_\tilde{r} {e_a}^\tilde{p} \partial_\tilde{b} \partial_\tilde{p}({e_d}^\tilde{r})
)
$
$
+ {\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{u}} {\tilde{\Gamma}^\tilde{u}}_{\tilde{b}\tilde{d}}
- {\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{u}} {\tilde{\Gamma}^\tilde{u}}_{\tilde{a}\tilde{d}}
$
$
+ \partial_\tilde{a}({e_u}^\tilde{c}) {\tilde{\Gamma}^u}_{\tilde{b}\tilde{d}}
- \partial_\tilde{b}({e_u}^\tilde{c}) {\tilde{\Gamma}^u}_{\tilde{a}\tilde{d}}
$
$
+ {\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{v}} {e^d}_\tilde{d} \partial_\tilde{b}({e_d}^\tilde{v})
- {\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{v}} {e^d}_\tilde{d} \partial_\tilde{a}({e_d}^\tilde{v})
$
$
+ \partial_\tilde{a}({e_u}^\tilde{c}) {e^u}_\tilde{v} {e^d}_\tilde{d} \partial_\tilde{b}({e_d}^\tilde{v})
- \partial_\tilde{b}({e_u}^\tilde{c}) {e^u}_\tilde{v} {e^d}_\tilde{d} \partial_\tilde{a}({e_d}^\tilde{v})
$
$
- {\tilde{\Gamma}^\tilde{c}}_{\tilde{u}\tilde{d}} {c_{\tilde{a}\tilde{b}}}^\tilde{u}
- {e^d}_\tilde{d} \partial_\tilde{u}({e_d}^\tilde{c}) {c_{\tilde{a}\tilde{b}}}^\tilde{u}
$
${R^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}} =
{e_c}^\tilde{c} \partial_\tilde{a}({e^c}_\tilde{r}) {\tilde{\Gamma}^\tilde{r}}_{\tilde{b}\tilde{d}}
+ {e^b}_\tilde{b} \partial_\tilde{a}({e_b}^\tilde{q}) {\tilde{\Gamma}^\tilde{c}}_{\tilde{q}\tilde{d}}
+ {e^d}_\tilde{d} \partial_\tilde{a}({e_d}^\tilde{s}) {\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{s}}
+ \partial_\tilde{a}({\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{d}})
$
$
- {e_c}^\tilde{c} \partial_\tilde{b}({e^c}_\tilde{r}) {\tilde{\Gamma}^\tilde{r}}_{\tilde{a}\tilde{d}}
- {e^a}_\tilde{a} \partial_\tilde{b}({e_a}^\tilde{p}) {\tilde{\Gamma}^\tilde{c}}_{\tilde{p}\tilde{d}}
- {e^d}_\tilde{d} \partial_\tilde{b} {e_d}^\tilde{s} {\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{s}}
- \partial_\tilde{b}({\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{d}})
$
$
+ {e^d}_\tilde{d} {e_c}^\tilde{c} \partial_\tilde{a}({e^c}_\tilde{r}) \partial_\tilde{b}({e_d}^\tilde{r})
+ {e^d}_\tilde{d} {e^b}_\tilde{b} \partial_\tilde{a}({e_b}^\tilde{q}) \partial_\tilde{q}({e_d}^\tilde{c})
+ {e^d}_\tilde{d} \partial_\tilde{a} \partial_\tilde{b}({e_d}^\tilde{c})
$
$
- {e^d}_\tilde{d} {e_c}^\tilde{c} \partial_\tilde{b}({e^c}_\tilde{r}) \partial_\tilde{a}({e_d}^\tilde{r})
- {e^d}_\tilde{d} {e^a}_\tilde{a} \partial_\tilde{b}({e_a}^\tilde{p}) \partial_\tilde{p}({e_d}^\tilde{c})
- {e^d}_\tilde{d} \partial_\tilde{b} \partial_\tilde{a}({e_d}^\tilde{c})
$
$
+ {\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{u}} {\tilde{\Gamma}^\tilde{u}}_{\tilde{b}\tilde{d}}
- {\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{u}} {\tilde{\Gamma}^\tilde{u}}_{\tilde{a}\tilde{d}}
$
$
+ \partial_\tilde{a}({e_u}^\tilde{c}) {\tilde{\Gamma}^u}_{\tilde{b}\tilde{d}}
- \partial_\tilde{b}({e_u}^\tilde{c}) {\tilde{\Gamma}^u}_{\tilde{a}\tilde{d}}
$
$
+ {\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{v}} {e^d}_\tilde{d} \partial_\tilde{b}({e_d}^\tilde{v})
- {\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{v}} {e^d}_\tilde{d} \partial_\tilde{a}({e_d}^\tilde{v})
$
$
+ \partial_\tilde{a}({e_u}^\tilde{c}) {e^u}_\tilde{v} {e^d}_\tilde{d} \partial_\tilde{b}({e_d}^\tilde{v})
- \partial_\tilde{b}({e_u}^\tilde{c}) {e^u}_\tilde{v} {e^d}_\tilde{d} \partial_\tilde{a}({e_d}^\tilde{v})
$
$
- {\tilde{\Gamma}^\tilde{c}}_{\tilde{u}\tilde{d}} {c_{\tilde{a}\tilde{b}}}^\tilde{u}
- {e^d}_\tilde{d} \partial_\tilde{u}({e_d}^\tilde{c}) {c_{\tilde{a}\tilde{b}}}^\tilde{u}
$
... using ${c_{\tilde{a}\tilde{b}}}^\tilde{c} = {e^b}_\tilde{b} \partial_\tilde{a} ({e_b}^\tilde{c}) - {e^a}_\tilde{a} \partial_\tilde{b} ({e_a}^\tilde{c})$...
${R^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}} =
\partial_\tilde{a}({\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{d}})
- \partial_\tilde{b}({\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{d}})
+ {\tilde{\Gamma}^\tilde{c}}_{\tilde{a}\tilde{u}} {\tilde{\Gamma}^\tilde{u}}_{\tilde{b}\tilde{d}}
- {\tilde{\Gamma}^\tilde{c}}_{\tilde{b}\tilde{u}} {\tilde{\Gamma}^\tilde{u}}_{\tilde{a}\tilde{d}}
$
$
+ {e^d}_\tilde{d} {e_c}^\tilde{c} \partial_\tilde{a}({e^c}_\tilde{r}) \partial_\tilde{b}({e_d}^\tilde{r})
- {e^d}_\tilde{d} {e_c}^\tilde{c} \partial_\tilde{a}({e_d}^\tilde{r}) \partial_\tilde{b}({e^c}_\tilde{r})
+ {e^u}_\tilde{v} {e^d}_\tilde{d} \partial_\tilde{a}({e_u}^\tilde{c}) \partial_\tilde{b}({e_d}^\tilde{v})
- {e^u}_\tilde{v} {e^d}_\tilde{d} \partial_\tilde{a}({e_d}^\tilde{v}) \partial_\tilde{b}({e_u}^\tilde{c})
$
$
+ {e^d}_\tilde{d} {e^b}_\tilde{b} \partial_\tilde{a}({e_b}^\tilde{u}) \partial_\tilde{u}({e_d}^\tilde{c})
- {e^d}_\tilde{d} {e^a}_\tilde{a} \partial_\tilde{u}({e_d}^\tilde{c}) \partial_\tilde{b}({e_a}^\tilde{u})
- {e^d}_\tilde{d} \partial_\tilde{u}({e_d}^\tilde{c}) {c_{\tilde{a}\tilde{b}}}^\tilde{u}
$
${R^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}} = {\tilde{R}^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}}
+ {e^d}_\tilde{d} {e_c}^\tilde{c} \partial_\tilde{a}({e^c}_\tilde{r}) \partial_\tilde{b}({e_d}^\tilde{r})
- {e^d}_\tilde{d} {e_c}^\tilde{c} \partial_\tilde{a}({e_d}^\tilde{r}) \partial_\tilde{b}({e^c}_\tilde{r})
+ {e^u}_\tilde{v} {e^d}_\tilde{d} \partial_\tilde{a}({e_u}^\tilde{c}) \partial_\tilde{b}({e_d}^\tilde{v})
- {e^u}_\tilde{v} {e^d}_\tilde{d} \partial_\tilde{a}({e_d}^\tilde{v}) \partial_\tilde{b}({e_u}^\tilde{c})
$
${R^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}} = {\tilde{R}^\tilde{c}}_{\tilde{d}\tilde{a}\tilde{b}}$
Therefore the curvature of the torsion-free geodesic is the same regardless of basis.
Trace of Riemann tensor over 1st and 2nd indexes (proof of their antisymmetry):
${R^u}_{uab} =
2 e_{[a} ({\Gamma^u}_{b]u})
+ 2 {\Gamma^u}_{[a|v} {\Gamma^v}_{b]u}
- {\Gamma^u}_{vu} {c_{ab}}^v
$
$= 2 e_{[a} ({\hat\Gamma^u}_{b]u} + {K^u}_{b]u})
+ {\Gamma^u}_{av} {\Gamma^v}_{bu}
- {\Gamma^u}_{bv} {\Gamma^v}_{au}
- ({\hat\Gamma^u}_{vu} + {K^u}_{vu})) {c_{ab}}^v
$
$= 2 e_{[a} ( e_{b]} ( log \sqrt{|g|} ) )
- e_v ( log \sqrt{|g|} ) {c_{ab}}^v
$
$= {c_{ab}}^v (e_v ( log \sqrt{|g|} ) - e_v ( log \sqrt{|g|}))$
$= 0$
Therefore $g^{ab} R_{abcd} = 0$
Does this necessarily imply that $h^{ab} R_{abcd} = 0$ for any symmetric $h^{ab}$?
(That is what is necessary to prove that $R_{abcd} = R_{[ab]cd}$)
Covariant derivative of (p,q) tensor (TODO put this in covariant derivative)
$\nabla_u (z) = \nabla_u ({z^A}_B {e_A}^B)$
$= e_u({z^A}_B) {e_A}^B + {z^A}_B \nabla_u ({e_A}^B)$
$= e_u({z^A}_B) {e_A}^B + {z^A}_B (
\underset{a_i \leftrightarrow c}{{e_A}^B} {\Gamma^c}_{u a_i}
- \underset{b_j \leftrightarrow c}{{e_A}^B} {\Gamma^{b_j}}_{u c}
)$
$= \left(
e_u({z^A}_B)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{uc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{u b_j}
\right) {e_A}^B$
Riemann curvature of (p,q) tensor:
$R(x,y) z = \nabla_x \nabla_y z - \nabla_y \nabla_x z - \nabla_{[x,y]} z$
$= x^a \nabla_a (y^u \nabla_u (z)) - y^a \nabla_a (x^u \nabla_u (z)) - (x^a e_a (y^u) - y^a e_a (x^u) + x^a y^b {c_{ab}}^u) \nabla_u (z)$
$= x^a e_a (y^u) \nabla_u (z)
+ x^a y^u \nabla_a (\nabla_u (z))
- y^a e_a (x^u) \nabla_u (z)
- y^a x^u \nabla_a (\nabla_u(z))
- (x^a e_a (y^u) - y^a e_a (x^u) + x^a y^b {c_{ab}}^u) \nabla_u (z)$
$= x^a y^u \nabla_a (\nabla_u (z))
- y^a x^u \nabla_a (\nabla_u(z))
- x^a y^b {c_{ab}}^u \nabla_u (z)$
$= (x^u y^v - y^u x^v) (\nabla_u (\nabla_v (z)) - \frac{1}{2} {c_{uv}}^r \nabla_r (z))$
$= 2 x^{[u} y^{v]} \left(
\nabla_u \left(
\left(
e_v \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{vc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{v b_j}
\right) {e_A}^B
\right)
- \frac{1}{2} {c_{uv}}^r \left(
e_r \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{rc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{r b_j}
\right) {e_A}^B
\right)$
$= 2 x^{[u} y^{v]} \left(
\left(
e_u \left(
e_v \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{vc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{v b_j}
\right)
- \frac{1}{2} {c_{uv}}^r \left(
e_r \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{rc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{r b_j}
\right)
\right)
{e_A}^B
+ \left(
e_v \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{vc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{v b_j}
\right)
\nabla_u ({e_A}^B)
\right)$
$= 2 x^{[u} y^{v]} \left(
\left(
e_u \left(
e_v \left( {z^A}_B \right)
\right)
+ \underset{a_i \leftrightarrow c}{ e_u \left( {z^A}_B \right) }
{\Gamma^{a_i}}_{vc}
+ \underset{a_i \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^{a_i}}_{vc} \right)
- \underset{b_j \leftrightarrow c}{ e_u \left( {z^A}_B \right) }
{\Gamma^c}_{v b_j}
- \underset{b_j \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^c}_{v b_j} \right)
- \frac{1}{2} {c_{uv}}^r \left(
e_r \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{rc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{r b_j}
\right)
\right)
{e_A}^B
+ \left(
e_v \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{vc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{v b_j}
\right) \left(
\underset{a_k \leftrightarrow d}{{e_A}^B} {\Gamma^d}_{u a_k}
- \underset{b_l \leftrightarrow d}{{e_A}^B} {\Gamma^{b_l}}_{u d}
\right)
\right)$
Using $x^{[u} y^{v]} {c_{uv}}^r e_r (s) = x^u y^v {c_{[uv]}}^r e_r = x^u y^v {c_{uv}}^r e_r (s) = x^u y^v (e_u (e_v ( s )) - e_v (e_u( s))) = (x^u y^v - x^v y^u) e_u ( e_v ( s ))$
$= 2 x^{[u} y^{v]} \left(
\left(
\frac{1}{2} {c_{uv}}^r e_r ( {z^A}_B )
+ \underset{a_i \leftrightarrow c}{ e_u \left( {z^A}_B \right) }
{\Gamma^{a_i}}_{vc}
+ \underset{a_i \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^{a_i}}_{vc} \right)
- \underset{b_j \leftrightarrow c}{ e_u \left( {z^A}_B \right) }
{\Gamma^c}_{v b_j}
- \underset{b_j \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^c}_{v b_j} \right)
- \frac{1}{2} {c_{uv}}^r \left(
e_r \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{rc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{r b_j}
\right)
\right)
{e_A}^B
+ \left(
e_v \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{vc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{v b_j}
\right) \left(
\underset{a_k \leftrightarrow d}{{e_A}^B} {\Gamma^d}_{u a_k}
- \underset{b_l \leftrightarrow d}{{e_A}^B} {\Gamma^{b_l}}_{u d}
\right)
\right)$
$= 2 x^{[u} y^{v]} \left(
\left(
\underset{a_i \leftrightarrow c}{ e_u \left( {z^A}_B \right) }
{\Gamma^{a_i}}_{vc}
+ \underset{a_i \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^{a_i}}_{vc} \right)
- \underset{b_j \leftrightarrow c}{ e_u \left( {z^A}_B \right) }
{\Gamma^c}_{v b_j}
- \underset{b_j \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^c}_{v b_j} \right)
- \frac{1}{2} {c_{uv}}^r \left(
\underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{rc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{r b_j}
\right)
\right)
{e_A}^B
+ \left(
e_v \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{vc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{v b_j}
\right) \left(
\underset{a_k \leftrightarrow d}{{e_A}^B} {\Gamma^d}_{u a_k}
- \underset{b_l \leftrightarrow d}{{e_A}^B} {\Gamma^{b_l}}_{u d}
\right)
\right)$
...
sidebar:
$x^{[u} y^{v]}
\left(
e_v \left( {z^A}_B \right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{vc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{v b_j}
\right) \left(
\underset{a_k \leftrightarrow d}{{e_A}^B} {\Gamma^d}_{u a_k}
- \underset{b_l \leftrightarrow d}{{e_A}^B} {\Gamma^{b_l}}_{u d}
\right)$
$= x^{[u} y^{v]}
\left(
e_v \left( {z^A}_B \right)
\left(
\underset{a_k \leftrightarrow d}{{e_A}^B} {\Gamma^d}_{u a_k}
- \underset{b_l \leftrightarrow d}{{e_A}^B} {\Gamma^{b_l}}_{u d}
\right)
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{vc}
\left(
\underset{a_k \leftrightarrow d}{{e_A}^B} {\Gamma^d}_{u a_k}
- \underset{b_l \leftrightarrow d}{{e_A}^B} {\Gamma^{b_l}}_{u d}
\right)
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{v b_j}
\left(
\underset{a_k \leftrightarrow d}{{e_A}^B} {\Gamma^d}_{u a_k}
- \underset{b_l \leftrightarrow d}{{e_A}^B} {\Gamma^{b_l}}_{u d}
\right)
\right) $
$= x^{[u} y^{v]}
\left(
e_v \left( {z^A}_B \right)
\underset{a_k \leftrightarrow d}{{e_A}^B} {\Gamma^d}_{u a_k}
- e_v \left( {z^A}_B \right)
\underset{b_l \leftrightarrow d}{{e_A}^B} {\Gamma^{b_l}}_{u d}
+ \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{vc}
\underset{a_k \leftrightarrow d}{{e_A}^B} {\Gamma^d}_{u a_k}
- \underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{vc}
\underset{b_l \leftrightarrow d}{{e_A}^B} {\Gamma^{b_l}}_{u d}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{v b_j}
\underset{a_k \leftrightarrow d}{{e_A}^B} {\Gamma^d}_{u a_k}
+ \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{v b_j}
\underset{b_l \leftrightarrow d}{{e_A}^B} {\Gamma^{b_l}}_{u d}
\right)
$
$= x^{[u} y^{v]}
\left(
\underset{a_k \leftrightarrow d}{e_v \left( {z^A}_B \right)}
{\Gamma^{a_k}}_{u d}
-
\underset{b_l \leftrightarrow d}{e_v \left( {z^A}_B \right)}
{\Gamma^d}_{u b_l}
+
\underset{a_k \leftrightarrow d}{
\underset{a_i \leftrightarrow c}{
{z^A}_B
}
}
{\Gamma^{a_i}}_{vc}
{\Gamma^{a_k}}_{u d}
-
\underset{b_l \leftrightarrow d}{
\underset{a_i \leftrightarrow c}{
{z^A}_B
}
}
{\Gamma^{a_i}}_{vc}
{\Gamma^d}_{u b_l}
-
\underset{a_k \leftrightarrow d}{
\underset{b_j \leftrightarrow c}{
{z^A}_B
}
}
{\Gamma^c}_{v b_j}
{\Gamma^{a_k}}_{u d}
+
\underset{b_l \leftrightarrow d}{
\underset{b_j \leftrightarrow c}{
{z^A}_B
}
}
{\Gamma^c}_{v b_j}
{\Gamma^d}_{u b_l}
\right)
{e_A}^B
$
cancel symmetric times antisymmetric sums...
$= 2 x^{[u} y^{v]}
\left(
\underset{a_k \leftrightarrow d}{e_v \left( {z^A}_B \right)}
{\Gamma^{a_k}}_{u d}
- \underset{b_l \leftrightarrow d}{e_v \left( {z^A}_B \right)}
{\Gamma^d}_{u b_l}
- \underset{b_l \leftrightarrow d}{
\underset{a_i \leftrightarrow c}{
{z^A}_B
}
}
{\Gamma^{a_i}}_{vc}
{\Gamma^d}_{u b_l}
- \underset{b_j \leftrightarrow c}{
\underset{a_k \leftrightarrow d}{
{z^A}_B
}
}
{\Gamma^c}_{v b_j}
{\Gamma^{a_k}}_{u d}
\right)
{e_A}^B
$
...
$= 2 x^{[u} y^{v]}
\left(
\underset{a_i \leftrightarrow c}{ e_u \left( {z^A}_B \right) }
{\Gamma^{a_i}}_{vc}
+ \underset{a_i \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^{a_i}}_{vc} \right)
- \underset{b_j \leftrightarrow c}{ e_u \left( {z^A}_B \right) }
{\Gamma^c}_{v b_j}
- \underset{b_j \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^c}_{v b_j} \right)
- \frac{1}{2} {c_{uv}}^r \left(
\underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{rc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{r b_j}
\right)
+ \underset{a_k \leftrightarrow d}{e_v \left( {z^A}_B \right)}
{\Gamma^{a_k}}_{u d}
- \underset{b_l \leftrightarrow d}{e_v \left( {z^A}_B \right)}
{\Gamma^d}_{u b_l}
- \underset{b_l \leftrightarrow d}{
\underset{a_i \leftrightarrow c}{
{z^A}_B
}
}
{\Gamma^{a_i}}_{vc}
{\Gamma^d}_{u b_l}
- \underset{b_j \leftrightarrow c}{
\underset{a_k \leftrightarrow d}{
{z^A}_B
}
}
{\Gamma^c}_{v b_j}
{\Gamma^{a_k}}_{u d}
\right)
{e_A}^B
$
using
$2 x^{[u} y^{v]}
\left(
\underset{a_i \leftrightarrow c}{
e_u \left( {z^A}_B \right)
}
{\Gamma^{a_i}}_{vc}
+ \underset{a_i \leftrightarrow c}{
e_v \left( {z^A}_B \right)
}
{\Gamma^{a_i}}_{u c}
\right) = 0$
and
$2 x^{[u} y^{v]}
\left(
\underset{b_j \leftrightarrow c}{
e_u \left( {z^A}_B \right)
}
{\Gamma^c}_{v b_j}
+ \underset{b_j \leftrightarrow c}{
e_v \left( {z^A}_B \right)
}
{\Gamma^c}_{u b_j}
\right) = 0 $,
since both are antisymmetric times symmetric
$= 2 x^{[u} y^{v]}
\left(
\underset{a_i \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^{a_i}}_{vc} \right)
- \underset{b_j \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^c}_{v b_j} \right)
- \underset{b_l \leftrightarrow d}{
\underset{a_i \leftrightarrow c}{
{z^A}_B
}
}
{\Gamma^{a_i}}_{vc}
{\Gamma^d}_{u b_l}
- \underset{b_j \leftrightarrow c}{
\underset{a_k \leftrightarrow d}{
{z^A}_B
}
}
{\Gamma^c}_{v b_j}
{\Gamma^{a_k}}_{u d}
- \frac{1}{2} {c_{uv}}^r \left(
\underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{rc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{r b_j}
\right)
\right)
{e_A}^B
$
$= 2 x^{[u} y^{v]}
\left(
\underset{a_i \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^{a_i}}_{vc} \right)
- \underset{b_j \leftrightarrow c}{{z^A}_B}
e_u \left( {\Gamma^c}_{v b_j} \right)
- \underset{b_l \leftrightarrow d}{
\underset{a_i \leftrightarrow c}{
{z^A}_B
}
}
{\Gamma^{a_i}}_{vc}
{\Gamma^d}_{u b_l}
- \underset{b_j \leftrightarrow c}{
\underset{a_k \leftrightarrow d}{
{z^A}_B
}
}
{\Gamma^c}_{v b_j}
{\Gamma^{a_k}}_{u d}
- \frac{1}{2} {c_{uv}}^r \left(
\underset{a_i \leftrightarrow c}{{z^A}_B} {\Gamma^{a_i}}_{rc}
- \underset{b_j \leftrightarrow c}{{z^A}_B} {\Gamma^c}_{r b_j}
\right)
\right)
{e_A}^B
$
$= x^u y^v
\left(
\underset{a_i \leftrightarrow c}{{z^A}_B}
\left(
2 e_{[u} \left( {\Gamma^{a_i}}_{v]c} \right)
- {\Gamma^{a_i}}_{rc}
{c_{uv}}^r
\right)
-
\underset{b_j \leftrightarrow c}{{z^A}_B}
\left(
2
e_{[u} \left( {\Gamma^c}_{v] b_j} \right)
-
{\Gamma^c}_{r b_j}
{c_{uv}}^r
\right)
+ 2 \underset{b_l \leftrightarrow d}{
\underset{a_i \leftrightarrow c}{
{z^A}_B
}
}
{\Gamma^{a_i}}_{[u| c}
{\Gamma^d}_{v] b_l}
- 2 \underset{b_j \leftrightarrow c}{
\underset{a_k \leftrightarrow d}{
{z^A}_B
}
}
{\Gamma^{a_k}}_{[u| d}
{\Gamma^c}_{v] b_j}
\right)
{e_A}^B
$
...
The goal should look like:
$= x^u y^v
\left(
\underset{a_i \leftrightarrow c}{ {z^A}_B }
\left(
2 e_{[u} ({\Gamma^{a_i}}_{v] c})
+ 2 {\Gamma^{a_i}}_{[u|r} {\Gamma^r}_{v] c}
- {\Gamma^{a_i}}_{r c} {c_{uv}}^r
\right)
- \underset{b_j \leftrightarrow c}{ {z^A}_B }
\left(
2 e_{[u} ({\Gamma^c}_{v]b_j})
+ 2 {\Gamma^c}_{[u|r} {\Gamma^r}_{v]b_j}
- {\Gamma^c}_{r b_j} {c_{uv}}^r
\right)
\right) {e_A}^B$
$= x^u y^v
\left(
\underset{a_i \leftrightarrow c}{ {z^A}_B } {R^{a_i}}_{c uv}
- \underset{b_j \leftrightarrow c}{ {z^A}_B } {R^c}_{b_j uv}
\right) {e_A}^B$
Asymmetry of covariant derivatives on a vector, with torsion. TODO reconcile this with the first identity:
$[\nabla_a, \nabla_b] v^c = 2 \nabla_{[a} \nabla_{b]} v^c$
$= 2 \nabla_{[a} ( e_{[b} ( v^c ) + {\Gamma^c}_{b]d} v^d )$
$= 2 ( \nabla_{[a} e_{[b} ( v^c )
+ \nabla_{[a} {\Gamma^c}_{b]d} v^d
+ {\Gamma^c}_{b]d} \nabla_{[a} v^d
)$
$= 2 (
e_{[a} ( e_{b]} ( v^c ) )
+ {\Gamma^c}_{[a|d} e_{b]} ( v^d )
- {\Gamma^d}_{[ab]} e_d ( v^c )
+ e_{[a} ( {\Gamma^c}_{b]d} ) v^d
+ {\Gamma^c}_{[a|u} {\Gamma^u}_{b]d} v^d
- {\Gamma^u}_{[ab]} {\Gamma^c}_{ud} v^d
- {\Gamma^u}_{[a|d} {\Gamma^c}_{b]u} v^d
+ {\Gamma^c}_{[b|d} e_{a]} v^d
+ {\Gamma^c}_{[b|d} {\Gamma^d}_{a]u} v^u
)$
$= 2 e_{[a} ( {\Gamma^c}_{b]d} ) v^d
+ 2 {\Gamma^c}_{[a|u} {\Gamma^u}_{b]d} v^d
- {\Gamma^c}_{ud} ({T^u}_{ab} + {c_{ab}}^u) v^d
+ 2 (
e_{[a} ( e_{b]} ( v^c ) )
- {\Gamma^d}_{[ab]} e_d ( v^c )
)
$
$= (
2 e_{[a} ( {\Gamma^c}_{b]d} )
+ 2 {\Gamma^c}_{[a|u} {\Gamma^u}_{b]d}
- {\Gamma^c}_{ud} {c_{ab}}^u
) v^d
- {T^u}_{ab} {\Gamma^c}_{ud} v^d
+ {c_{ab}}^d e_d (v^c)
- {T^d}_{ab} e_d ( v^c )
- {c_{ab}}^d e_d ( v^c )
$
$= {R^c}_{dab} v^d
- {T^u}_{ab} (e_u ( v^c ) + {\Gamma^c}_{ud} v^d)
$
$= {R^c}_{dab} v^d
- {T^u}_{ab} \nabla_u v^c
$
TODO compare the Riemann curvature based on a non-Levi-Civita connection (with torsion) with a Riemann curvature of the Levi-Civita connection.
Riemann curvature two-form:
${\Omega^c}_d = {R^c}_{dab} e^a \otimes e^b$
$= \frac{1}{2} {R^c}_{dab} e^a \wedge e^b$
$= (
2 e_{[a} ({\Gamma^c}_{b]d})
+ 2 {\Gamma^c}_{[a|u} {\Gamma^u}_{b]d}
- {\Gamma^c}_{ud} {c_{ab}}^u
) e^a \otimes e^b$
$= (
e_{[a} ({\Gamma^c}_{b]d})
- \frac{1}{2} {\Gamma^c}_{ud} {c_{ab}}^u
+ {\Gamma^c}_{[a|u} {\Gamma^u}_{b]d}
) e^a \wedge e^b$
$ = d {\omega^c}_d + {\omega^c}_u \wedge {\omega^u}_d$
Since $e^a \wedge e^b$ is antisymmetric, we know ${R^c}_{dab} = {R^c}_{d[ab]}$.
Second exterior derivative of a basis vector:
$d e_u = e_v \wedge {\omega^v}_u$
$d^2 e_u = d e_v \wedge {\omega^v}_u + e_v \wedge d {\omega^v}_u$
$d^2 e_u = e_a \wedge {\omega^a}_v \wedge {\omega^v}_u + e_v \wedge d {\omega^v}_u$
$d^2 e_u = e_a \wedge (d {\omega^a}_u + {\omega^a}_v \wedge {\omega^v}_u)$
$d^2 e_u = e_a \wedge {\Omega^a}_u$
So can I say?...
$e^v(d^2 e_u) = e^v(e_a \wedge {\Omega^a}_u)$
...partial application...
$= \delta^v_a \wedge {\Omega^a}_u$
$= {\Omega^v}_u$
Or is it better notation to say?...
$d^2 e_v (e_a \otimes e_b) = e_u \wedge {\Omega^u}_v (e_a \otimes e_b)$
$= e_u \wedge {R^u}_{vcd} (e^c \otimes e^d) (e_a \otimes e_b)$
$= e_u \wedge {R^u}_{vab}$
Mind you since that $e_u$ is a vector and the ${R^u}_{vab}$ is a 0-form, is the $\wedge$ inter-operable with a $\otimes$?
Or is it better to say (in the 2-form basis)?:
$e_v \lrcorner e_u \lrcorner {\Omega^a}_{b}$
$= e_v \lrcorner e_u \lrcorner {R^a}_{bcd} e^c \otimes e^d$
$= e_v \lrcorner {R^a}_{bcd} \delta^c_u e^d$
$= {R^a}_{bcd} \delta^c_u \delta^d_v$
$= {R^a}_{buv}$
How about equivalently, can interior product be used for k-vectors?:
$(e_u \otimes e_v) \lrcorner {\Omega^a}_b$
$= (e_u \otimes e_v) \lrcorner {R^a}_{bcd} e^c \otimes e^d$
$= {R^a}_{bcd} \delta^c_u \delta^d_v$
$= {R^a}_{buv}$
Second exterior derivative of a vector:
$v = v^u e_u$
$v = e_u \wedge v^u$
$d v = d e_u \wedge v^u + e_u \wedge d v^u$
using $d e_u = e_v \wedge {\omega^v}_u$...
$d v = e_a \wedge {\omega^a}_u \wedge v^u + e_a \wedge d v^a$
$d v = e_a \wedge (d v^a + {\omega^a}_u \wedge v^u)$
$d v = e_a \wedge Dv^a$
$d^2 v = d e_a \wedge (d v^a + {\omega^a}_v \wedge v^v) + e_u \wedge (d^2 v^u + d ({\omega^u}_v \wedge v^v))$
using $d(a \wedge b) = da \wedge b + (-1)^p a \wedge db$, and that ${\omega^a}_b$ is a 1-form...
$d^2 v = d e_a \wedge (d v^a + {\omega^a}_v \wedge v^v) + e_u \wedge (d^2 v^u + d {\omega^u}_v \wedge v^v - {\omega^u}_v \wedge d v^v)$
$d^2 v = e_u \wedge {\omega^u}_a \wedge (d v^a + {\omega^a}_v \wedge v^v) + e_u \wedge (d^2 v^u + d {\omega^u}_v \wedge v^v - {\omega^u}_v \wedge d v^v)$
$d^2 v = e_u \wedge (
{\omega^u}_a \wedge d v^a
+ {\omega^u}_a \wedge {\omega^a}_v \wedge v^v
+ d^2 v^u
+ d {\omega^u}_v \wedge v^v
- {\omega^u}_v \wedge d v^v
)$
...using $d^2 v^u = 0$ since $v^u$ is a scalar i.e. a 0-form...
$d^2 v = e_u \wedge (d {\omega^u}_v + {\omega^u}_a \wedge {\omega^a}_v) \wedge v^v$
...using ${\Omega^u}_v = d {\omega^u}_v + {\omega^u}_a \wedge {\omega^a}_v$...
$d^2 v = e_u \wedge {\Omega^u}_v \wedge v^v$
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