The Gauss-Codazzi Equations

Gauss-Codazzi
The Gauss-Codazzi equations are fundamental equations in Riemaniann geometry, where they are used in the theory of embedded hypersurfaces in a Euclidean space. The first equation was derived by Gauss in 1828 (Gauss, 1828) and is the basis for Gauss’s “Theorema Egregium”, which states that the Gaussian curvature of a surface is invariant under local isometry. It relates the n-curvature in terms of the intrinsic (n-1)-geometry of the hypersurface. The second equation is named for Delfino Codazzi, although it was derived earlier by Peterson in 1853 (Peterson, 1853), and relates the n-curvature in terms of the extrinsic curvature (often referred to as the second fundamental form) of the hypersurface.
Foliation of space time
Derivation
For all the spacetime coordinates the greek indices are used, which run from 1 to d+1. When considering components of the first foliation, the roman indices are used which run from 1 to d.
Throughout this derivation we work with Euclidean signature. Covariant derivatives of vectors are given by ∇_μ V^ν=∂_μ V^ν+Γ_μλ^ν V^λ and for one-forms by ∇_μ A_ν=∂_μ A_ν+Γ_μν^λ A_λ. The convention for the Riemann tensor is
R_ρνσ^μ=∂_ν Γ_ρσ^μ+Γ_λν^μ Γ_ρσ^λ-(ν↔σ)
The Einstein tensor is G_μν=R_μν-1/2 g_μν R and the Einstein field equations are
G_μν=κ^2 T_μν with κ^2=8πG_(d+1) and T^μν≡2/√g (δS_matter)/(δg_μν ). We consider the standard Einstein-Hilbert action with the Gibbons-Hawking boundary term to ensure that the variational problem is well-defined:
S=S_gr+S_m
=-1/(2κ^2 ) [∫_M^ ▒〖d^(d+1) x √g〗 R+∫_∂M^ ▒〖d^d x √γ 〗 2K]+∫_M^ ▒〖d^(d+1) x √g〗 L_m with L_m a generic matter field Lagrangian density and K the trace of the extrinsic curvature of the boundary.
We start with a quick derivation of the Gaus...

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... with [∇_κ,∇_λ ] we obtain e_a^κ e_b^λ [∇_κ,∇_λ ] e_c^μ=(∂_a Γ_bc^d+Γ_bc^e Γ_ae^d-Γ_ab^e Γ_ce^d ) e_d^μ+(∂_a K_bc+Γ_bc^d K_ad-Γ_ab^d K_cd ) n^μ+K_bc e_a^κ ∇_κ n^μ-K_ab n^λ ∇_λ e_c^μ-[(∂_b Γ_ac^d+Γ_ac^e Γ_be^d-Γ_ba^e Γ_ce^d ) e_d^μ+(∂_b K_ac+Γ_ac^d K_bd-Γ_ba^d K_cd ) n^μ+K_ac e_b^κ ∇_κ n^μ-K_ba n^λ ∇_λ e_c^μ ]
=(∂_a Γ_bc^d+Γ_bc^e Γ_ae^d-∂_b Γ_ac^d-Γ_ac^e Γ_be^d-Γ_ab^e Γ_ce^d+Γ_ba^e Γ_ce^d ) e_d^μ+(∂_a K_bc+Γ_bc^d K_ad-Γ_ab^d K_cd-∂_b K_ac-Γ_ac^d K_bd+Γ_ba^d K_cd ) n^μ+K_bc e_a^κ ∇_κ n^μ-K_ac e_b^κ ∇_κ n^μ-K_ab n^λ ∇_λ e_c^μ+K_ba n^λ ∇_λ e_c^μ
=R_cab^d-(∇_b K_ac-∇_a K_bc ) n^μ-(K_ac e_b^κ ∇_κ-K_bc e_a^κ ∇_κ ) n^μ
Now we only need to replace the commutator with the curvature tensor and contract equation with e_eμ and n_μ respectively, to obtain e_a^μ e_b^ν e_c^ρ e_d^σ R ̂_μνρσ=R_abcd-(K_ad K_bc-K_ac K_bd) n^μ e_a^ν e_b^ρ e_c^σ R ̂_μνρσ=∇_b K_ac-∇_c K_ab