preview

The Gauss-Codazzi Equations Essay

Decent Essays
Open Document

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 …show more content…

These terms contracted with some generice vector V ̂^μ that is tangent to the hypersurface, yields n_μ V ̂^μ=0 and e_a^μ V^a=V ̂^μ Hence the derivative of For this derivation we need to define the normal and tangent vectors of our hypersurface. We obtain e_a^λ ∇_λ e_b^μ=Γ_ab^c e_c^μ+K_ab n^μ Now we can compute the Gauss-Codazzi equations. We do so by calculating the curvature tensor, which measures the noncommutativity of the covariant derivative. Hence we would like to obtain the expression for [∇_κ,∇_λ ] V ̂^α=R ̂_(βμν )^α V^β and [∇_a,∇_b ] V^c=R_abd^c V^d So we begin with taking the covariant derivative of the basis vector e_a^κ ∇_κ (e_b^λ ∇_λ e_c^μ )=e_a^κ ∇_κ (Γ_bc^d e_d^μ+K_bc n^μ ) =∂_a Γ_bc^d e_d^μ+Γ_bc^d (e_a^κ ∇_κ e_d^μ )+∂_a K_bc n^μ+K_bc e_a^κ ∇_κ n^μ =∂_a Γ_bc^d e_d^μ+Γ_bc^d (Γ_ad^e e_e^μ+K_abd n^μ)+∂_a K_bc n^μ+K_bc e_a^κ ∇_κ n^μ Rewriting the left hand side yields e_a^κ ∇_κ (e_b^λ ∇_λ e_c^μ )=e_a^κ e_b^λ ∇_κ ∇_λ e_c^μ+(e_a^κ ∇_κ e_b^λ ) ∇_λ e_c^μ =e_a^κ e_b^λ ∇_κ ∇_λ e_c^μ+(Γ_ab^d e_d^λ+K_ab n^λ ) ∇_λ e_c^μ =e_a^κ e_b^λ ∇_κ ∇_λ e_c^μ+Γ_ab^d (Γ_cd^e e_e^μ+K_cd n^μ )+K_ab n^λ ∇_λ e_c^μ Combining these two expressions gives us e_a^κ e_b^λ ∇_κ ∇_λ e_c^μ=(∂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 We are however interested in the commutator of the covariant derivatives. When replacing ∇_κ ∇_λ with [∇_κ,∇_λ ] we obtain e_a^κ e_b^λ [∇_κ,∇_λ

Get Access
Related
Get Access