Verifying identity involving covariant derivative

[demonelite123]

i am trying to verify the following identity:
0 = ∂g_mn / ∂y^p + Γ ^s _pm g_sn + Γ ^r _pn g_mr

where Γ is the christoffel symbol with ^ telling what is the upper index and _ telling what are the two lower indices. g_mn is the metric tensor with 2 lower indices and y^p is the component of y the partial derivative is being taken with respect to. basically this equality shows that the covariant derivative of the metric tensor is 0.

so i expanded the christoffel symbols out according to the definition and the g^sd included in the first christoffel symbol cancels with the g_sn multiplying the first christoffel symbol and i get δ ^d _n where δ is the kronecker delta with upper index d and lower index n. i do something similar for the second christoffel symbol. i use the fact that the metric tensor is symmetric and that the christoffel symbols are symmetric with respect to their lower indices so in the end i get:
0 = 2 (∂g_mn / ∂y^p) after the rest cancel out.

so what i am stuck on is how to show that the right side equals the left side. did i do something wrong?

 

[sgd37]

yes for a type (0,2) tensor i.e. all the indecies downstairs you subtract the Christoffel symbols not add them. You add them when you have upstairs indecies so your equation should be

0 = ∂g_mn / ∂y^p - Γ ^s _pm g_sn - Γ ^r _pn g_mr

I believe that takes care of your problem

 

[demonelite123]

yes for a type (0,2) tensor i.e. all the indecies downstairs you subtract the Christoffel symbols not add them. You add them when you have upstairs indecies so your equation should be

0 = ∂g_mn / ∂y^p - Γ ^s _pm g_sn - Γ ^r _pn g_mr

I believe that takes care of your problem

ah i see, thanks!