Gauss-Codazzi-Ricci Relations

The Gauss–Codazzi–Ricci relations are compatibility identities for an embedded submanifold. For a hypersurface, they split ambient curvature into a fully tangential part controlled by intrinsic and extrinsic curvature, a part with one normal controlled by derivatives of the extrinsic curvature, and a normal-evolution part. They are geometric identities, not gravitational field equations.

Because every sign follows from the normal norm, extrinsic-curvature convention, and curvature commutator, those choices are recorded explicitly and then tested in both Riemannian and Lorentzian flat embeddings.

Their real content is integrability. Induced and extrinsic geometry cannot be prescribed independently if they are to arise from one ambient connection. The relations expose the combinations that must agree, and simple flat embeddings provide strong checks because every nonzero intrinsic term must then be balanced entirely by the bending data.

Required background. Affine Connections and Covariant Derivatives supplies the covariant derivative that is projected onto the hypersurface; The Riemann Curvature Tensor fixes the curvature commutator and index order; Pseudo-Riemannian Metrics and Signature supplies non-null normals and induced signatures. Helpful background. Calculus and Linear Algebra for Relativity supplies multilinear projections; Exterior Algebra and Differential Forms supplies orientation and antisymmetrization.

Normal decomposition and shape map

Let $n^\mu n_\mu=\epsilon=\pm1$ and

$$h^\mu{}_\nu=\delta^\mu{}_\nu-\epsilon n^\mu n_\nu.$$

We use

$$K_{\mu\nu} =-h_\mu{}^\alpha h_\nu{}^\beta\nabla_\alpha n_\beta.$$

For tangent fields $X,Y$ the Gauss and Weingarten formulas are

$$\nabla_XY=D_XY+\epsilon K(X,Y)n, \qquad \nabla_Xn=-S(X),$$

with $K(X,Y)=h(SX,Y)$. The ambient and intrinsic curvature conventions are both defined by the same commutator order. These declarations determine every sign below.

The Gauss relation

Apply the ambient curvature operator to tangent $Z$:

$$R(X,Y)Z =\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z.$$

Insert the Gauss formula twice and take the tangential projection. Derivatives of $K$ multiply the normal and drop out, while differentiating the normal converts back through the shape map. The result is

$$h(R(X,Y)Z,W) =h({}^{(\Sigma)}R(X,Y)Z,W) +\epsilon\left[ K(X,Z)K(Y,W)-K(Y,Z)K(X,W) \right].$$

Equivalently in indices,

$$h_a{}^\alpha h_b{}^\beta h_c{}^\gamma h_d{}^\delta R_{\alpha\beta\gamma\delta} ={}^{(\Sigma)}R_{abcd} -\epsilon\left(K_{ac}K_{bd}-K_{ad}K_{bc}\right).$$

The apparent sign change between the two displays is only index order. With the site’s convention $R_{\rho\sigma\mu\nu}=g(R(\partial_\mu,\partial_\nu)\partial_\sigma,\partial_\rho)$, the indexed left-hand side is $h(R(e_c,e_d)e_b,e_a)$, not $h(R(e_a,e_b)e_c,e_d)$. Reordering the symmetric $K$ factors converts the plus-$\epsilon$ coordinate-free bracket into the minus-$\epsilon$ indexed bracket. The quadratic term is unchanged if the normal is reversed because it contains two factors of $K$; it does depend on $\epsilon$ and on the adopted curvature index order.

The Codazzi relation

Take the normal component of the same curvature calculation. The algebraic $K^2$ terms cancel, leaving

$$n_\rho R^\rho{}_{\sigma\mu\nu} h^\sigma{}_b h^\mu{}_c h^\nu{}_d =D_cK_{db}-D_dK_{cb}.$$

This form follows from the stated minus-sign definition of $K$. It says that the one-normal, three-tangent projection of ambient curvature measures the antisymmetrized intrinsic derivative of the second fundamental form.

Equivalently, after pair exchange and antisymmetry in the first index pair,

$$h_a{}^\alpha h_b{}^\beta h_c{}^\gamma n^\delta R_{\alpha\beta\gamma\delta} =D_bK_{ac}-D_aK_{bc}.$$

For spacelike slices, the Gauss relation and its contractions are developed in Gourgoulhon 2007, §2.5.1, Eq. (2.92), pp. 34–36, PDF, and Codazzi in Gourgoulhon 2007, §2.5.2, Eq. (2.101), p. 36, PDF. The formulas here retain $\epsilon$ so that timelike and spacelike non-null hypersurfaces are covered together.

Two normal directions and normal evolution

For codimension greater than one, the normal connection can have nontrivial curvature. The Ricci equation relates that curvature to the two-normal projection of ambient curvature and commutators of shape maps. A hypersurface has a one-dimensional normal bundle, so there is no independent continuous normal rotation and the normal-bundle curvature is trivial. The remaining two-normal curvature projection instead enters normal evolution equations; calling it a new intrinsic constraint would obscure its geometric role.

This remaining projection has a particularly transparent local form in Gaussian normal coordinates. On a regular tubular neighborhood, write

$$g=\epsilon\,ds^2+h_{ab}(s,y)dy^a dy^b, \qquad n=\partial_s, \qquad K_{ab}=-\frac12\partial_s h_{ab}.$$

The connection coefficients with normal indices are $\Gamma^s{}_{ab}=\epsilon K_{ab}$ and $\Gamma^a{}_{sb}=-K^a{}_b$. Substitution into the curvature definition gives

$$R_{s a s b}=\partial_s K_{ab}+K_{ac}K^c{}_b.$$

The derivative here acts on the two-lowered-index components in these coordinates. Raising an index before differentiating also differentiates $h^{ab}$ and changes the displayed quadratic term. The formula holds for either sign of $\epsilon$, with $s$ proper distance or proper time along the normal geodesics. The more general lapse-dependent spacelike-slice projection is derived in Gourgoulhon 2007, §3.4, Eq. (3.43), pp. 47–48, PDF.

For concentric spheres, $h=r^2\gamma$ and $K=-r\gamma$, so $\partial_rK=-\gamma$ cancels $K_{ac}K^c{}_b=\gamma$. The same cancellation holds for Milne slices with $r$ replaced by $\tau$. These are independent checks that this projection vanishes in flat ambient geometry despite nonzero extrinsic curvature.

Independent sign checks

For a flat hyperplane in flat ambient space, ambient curvature, intrinsic curvature, and $K$ all vanish. Every relation reduces to zero.

For a round two-sphere of radius $R$ in Euclidean three-space with outward normal, $\epsilon=+1$ and the convention gives $K_{ab}=-h_{ab}/R$. Ambient curvature vanishes, so Gauss gives

$${}^{(\Sigma)}R_{abcd} =K_{ac}K_{bd}-K_{ad}K_{bc}.$$

Its sectional curvature is $1/R^2$, as required. This check fixes the relative sign between the intrinsic curvature and $K^2$ terms.

For a constant-$\tau$ Milne slice in flat spacetime, $\epsilon=-1$ and $K_{ij}=-h_{ij}/\tau$. The induced metric is hyperbolic with sectional curvature $-1/\tau^2$. Substitution into Gauss again gives the correct sign while the ambient Riemann tensor remains zero. The sphere and Milne checks together test both values of $\epsilon$.

Scope and limitations

Contracting Gauss and Codazzi produces scalar and vector projection identities. In a theory with specified field equations, those identities can become constraints on initial data. No such substitution is made here: ambient Ricci components remain geometric curvature components, not matter densities or currents.

The relations are necessary compatibility conditions for embedding data. Sufficiency requires appropriate regularity, topology, and integrability hypotheses. Global embeddability cannot be inferred from a pointwise identity alone. Null hypersurfaces likewise require a separate degenerate formalism and are not obtained by taking $\epsilon$ to zero.

At corners or boundaries of a hypersurface, the local identities still hold on smooth pieces, but integration by parts introduces induced orientations and joint terms. Those bookkeeping questions do not modify the pointwise curvature decomposition; they determine how projected identities contribute to integral statements.

Exercises

Use the Gauss relation to find the intrinsic scalar curvature of a $d$-dimensional spacelike Milne slice with

$$K_{ab}=-\frac1\tau h_{ab}$$

in flat spacetime.

Solution

Here $\epsilon=-1$ and the ambient tangential curvature vanishes. Gauss gives

$${}^{(\Sigma)}R_{abcd} =-\frac1{\tau^2} \left(h_{ac}h_{bd}-h_{ad}h_{bc}\right).$$

Contracting the first and third, then the second and fourth indices, yields

$${}^{(\Sigma)}R =-\frac{d(d-1)}{\tau^2}.$$

Thus the slice has constant negative sectional curvature even though the ambient spacetime is flat.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, 2007. arXiv:gr-qc/0703035.