| Diffenrantial Geometry.....=∧=
Christoffel symbol
Γ_(σμν) first kind and Γ_(μν) ^( σ) second kind...
and Chritoffel symbol satisfy
Γ_(μν) ^( σ) =(1/2)g^(σl) Γ_(lμν)
Γ_(ijk) {θ,ρ}∈{i,j,k}
Γ_(θθθ) , Γ_(ρρρ) , Γ_(θθρ)
Γ_(θρθ) , Γ_(ρθρ) , Γ_(ρθθ)
Γ_(θρρ) , Γ_(ρρθ)
Γ_(jk) ^i =(1/2)g^(il) Γ_(ljk)
g_(μν) = ((( r^2 ),( 0)),(( 0),(r^2 sin^2 (θ))) )
(g_(μν) )^(−1) =g^(μν)
g^(μν) = ((( (1/r^2 )),( 0)),(( 0),(1/(r^2 sin^2 (θ)))) )
g_(θθ) =r^2 , g_(θρ) =0 , g_(ρθ) =0 , g_(ρρ) =r^2 sin^2 (θ)
∂_θ g_(θθ) =0 , ∂_ρ g_(θθ) =0 , ∂_θ g_(θρ) =0 , ∂_ρ g_(θρ) =0
∂_θ g_(ρθ) =0 , ∂_ρ g_(ρθ) =0 , ∂_θ g_(ρρ) =2r^2 sin(θ)cos(θ) , ∂_ρ g_(ρρ) =0
Γ_(abc) =(1/2)(∂_b ^ g_(ac) +∂_c g_(ab) −∂_a g_(bc) )
Γ_(θθθ) =(1/2)(∂_θ g_(θθ) +∂_θ g_(θθ) −∂_θ g_(θθ) )=0
Γ_(ρρρ) =(1/2)(∂_ρ g_(ρρ) +∂_ρ g_(ρρ) −∂_ρ g_(ρρ) )=0
Γ_(θρρ) =(1/2)(∂_ρ g_(θρ) +∂_ρ g_(θρ) −∂_θ g_(ρρ) )=−r^2 sin(θ)cos(θ)
Γ_(ρθρ) =(1/2)(∂_θ g_(ρρ) +∂_ρ g_(ρθ) −∂_ρ g_(θρ) )=r^2 sin(θ)cos(θ)
Γ_(ρθθ) =(1/2)(∂_θ g_(ρθ) +∂_θ g_(ρθ) −∂_ρ g_(θθ) )=0
Γ_(θρθ) =(1/2)(∂_ρ g_(θθ) +∂_θ g_(θρ) −∂_θ g_(ρθ) )=0
Γ_(θθρ) =(1/2)(∂_θ g_(θρ) +∂_ρ g_(θθ) −∂_θ g_(θρ) )=0
Γ_(ρρθ) =(1/2)(∂_ρ g_(ρθ) +∂_θ g_(ρρ) −∂_ρ g_(ρθ) )=r^2 sin(θ)cos(θ)
Γ_(jk) ^i =g^(il) Γ_(ljk)
g^(θθ) =(1/r^2 ) , g^(θρ) =0 , g^(ρθ) =0 , g^(ρρ) =(1/(r^2 sin^2 (θ)))
Γ_(θθ) ^θ =g^(θl) Γ_(lθθ) = { ((g^(θθ) Γ_(θθθ) =0)),((g^(θρ) Γ_(ρθθ) =0)) :}, Γ_(ρθ) ^θ =g^(θl) Γ_(lρθ) = { ((g^(θθ) Γ_(θρθ) =0)),((g^(θρ) Γ_(ρρθ) =0)) :} , Γ_(θρ) ^θ =g^(θl) Γ_(lθρ) = { ((g^(θθ) Γ_(θθρ) =0)),((g^(θρ) Γ_(ρθρ) =0)) :} , Γ_(ρρ) ^θ =g^(θl) Γ_(lρρ) = { ((g^(θθ) Γ_(θρρ) =−sin(θ)cos(θ))),((g^(θρ) Γ_(ρρρ) =0)) :}
Γ_(θθ) ^ρ =g^(ρl) Γ_(lθθ) = { ((g^(ρθ) Γ_(θθθ) =0)),((g^(ρρ) Γ_(ρθθ) =0)) :} , Γ_(ρθ) ^ρ =g^(ρl) Γ_(lρθ) = { ((g^(ρθ) Γ_(θρθ) =0)),((g^(ρρ) Γ_(ρρθ) =cot(θ))) :}, Γ_(θρ) ^ρ =g^(ρl) Γ_(lθρ) = { ((g^(ρθ) Γ_(θθρ) =0)),((g^(ρρ) Γ_(ρθρ) =cot(θ))) :} , Γ_(ρρ) ^ρ =g^(ρl) Γ_(lρρ) = { ((g^(ρθ) Γ_(θρρ) =0)),((g^(ρρ) Γ_(ρρρ) =0)) :}
∴Γ_(ρρ) ^θ =−sin(θ)cos(θ)
Γ_(μν) ^( ρ) =Γ_(νμ) ^ρ
Γ_(ρθ) ^ρ =Γ_(θρ) ^ρ =cot(θ)
R_(jkl) ^i =∂_k Γ_(jl) ^k −∂_l Γ_(jk) ^i +Γ_(km) ^i Γ_(jl) ^m −Γ_(lm) ^i Γ_(jk) ^m
R_(θθθ) ^θ , R_(θρθ) ^θ , R_(ρθθ) ^θ , R_(ρρθ) ^θ , R_(θρρ) ^θ , R_(ρρρ) ^θ , R_(ρθρ) ^θ , R_(θρρ) ^θ
R_(θθθ) ^ρ , R_(θρθ) ^ρ , R_(ρθθ) ^ρ , R_(ρρθ) ^ρ , R_(θρρ) ^ρ , R_(ρρρ) ^ρ , R_(ρθρ) ^ρ , R_(θρρ) ^ρ
and Riemann metric tensor have symmetries
R_(abcd) =−R_(bacd)
R_(abcd) =−R_(abdc)
R_(abcd) =R_(cdab)
Non-Zero Γ_(μν) ^( ρ)
Γ_(ρρ) ^( θ) =−sin(θ)cos(θ) , Γ_(ρθ) ^( ρ) , Γ_(θρ) ^( ρ) =cot(θ)
{i,j,k,ℓ}∈{θ,ρ}
R_(ρθρ) ^θ =∂_θ Γ_(ρρ) ^θ −∂_ρ Γ_(ρθ) ^θ +Γ_(θm) ^( θ) Γ_(ρρ) ^m −Γ_(ρm) ^θ Γ_(ρθ) ^m
=sin^2 (θ)
R_(αρθρ) =g_(αμ) R_(ρθρ) ^μ
R_(θρθρ) =g_(θμ) R_(ρθρ) ^μ = { ((g_(θθ) R_(ρθρ) ^θ =r^2 sin^2 (θ))),((g_(θρ) R_(ρθρ) ^ρ =0)) :}
R_(θρθρ) =r^2 sin^2 (θ)
∴ R_(ρθρ) ^θ =sin^2 (θ) , R_(θρθρ) =r^2 sin^2 (θ)
∼2-dimensional Riemann Manifold∼
R_(abcd) =K(g_(ac) g_(bd) −g_(ad) g_(bc) )
R_(μν) ^(Ricci) =Kg_(μν)
K is gaussian curvature aka K=((detII_p )/(det I_p ))=((LM−N^2 )/(EG−F^2 ))
Q225479
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