Gauss-Codazzi-Mainardi Relations

The Gauss–Codazzi–Mainardi identities split ambient curvature into tangential, one-normal, and two-normal projections relative to a non-null hypersurface. This record fixes the full sign package and gives the contractions most often needed in geometric calculations. The longer construction and embedding interpretation are developed in Hypersurfaces, Extrinsic Curvature, and Embeddings and its Gauss–Codazzi–Ricci derivation.

Helpful background. Tensor Fields and Natural Operations supplies abstract-index projections; Affine Connections and Covariant Derivatives supplies the induced derivative and curvature commutator.

Non-null hypersurface conventions

Let $n^a$ be a chosen unit normal to a spacelike or timelike hypersurface $\Sigma$:

$$n^a n_a=\epsilon, \qquad \epsilon=\pm1, \qquad h^a{}_b=\delta^a{}_b-\epsilon n^a n_b.$$

The induced metric is $h_{ab}=g_{ab}-\epsilon n_a n_b$, and $D_a$ is its Levi–Civita derivative. Use the site curvature convention

$$[\nabla_c,\nabla_d]V^a=R^a{}_{bcd}V^b, \qquad R_{bd}=R^a{}_{bad},$$

together with

$$K_{ab}=-h_a{}^c h_b{}^d\nabla_c n_d, \qquad a_b=n^c\nabla_c n_b.$$

Thus $K_{ab}$ and $a_b$ are tangential, and

$$\nabla_a n_b=-K_{ab}+\epsilon n_a a_b.$$

The hypotheses are a smooth embedded hypersurface, a smooth unit normal, and the ambient Levi–Civita connection. Metric compatibility and hypersurface orthogonality make $K_{ab}$ symmetric. Indices on tangential tensors are raised with $h^{ab}$; writing an ambient metric instead gives the same result only after all free slots have been projected. The value $\epsilon=-1$ describes a spacelike hypersurface with timelike normal, while $\epsilon=+1$ describes a timelike hypersurface in Lorentzian spacetime or a hypersurface in a Riemannian ambient space.

The formulas are dimension-independent for non-null hypersurfaces. No orientation is needed beyond choosing the normal; changing coorientation sends $n\mapsto-n$ and $K\mapsto-K$ consistently.

These declarations translate Gourgoulhon’s spacelike-slice formulas as one package and agree with the pseudo-Riemannian submanifold construction in Gourgoulhon 2007, §§2.2–2.5, printed pp. 15–38, PDF and Wendl 2021–2022, §28.1, Proposition 28.5, printed pp. 238–241, Open PDF.

Tangential and one-normal projections

With every displayed free index tangential, the Gauss equation is

$$h_a{}^e h_b{}^f h_c{}^g h_d{}^k R_{efgk} ={}^{(h)}R_{abcd} -\epsilon\left(K_{ac}K_{bd}-K_{ad}K_{bc}\right).$$

The normal-last Codazzi–Mainardi equation is

$$h_a{}^e h_b{}^f h_c{}^g n^k R_{efgk} =D_bK_{ac}-D_aK_{bc}.$$

The position of the normal matters: moving it across an antisymmetric Riemann pair changes the sign. Reversing $n^a$ changes both $K_{ab}$ and the one-normal projection, so the identity remains invariant. These component orders follow Gourgoulhon 2007, Eqs. (2.92) and (2.101), printed pp. 35–36, PDF.

Gauss compares curvature calculated solely from $h_{ab}$ with the tangential ambient projection and the quadratic bending term. Codazzi instead detects whether tangential variations of $K_{ab}$ are compatible with the mixed ambient curvature. Neither formula reconstructs the ambient Riemann tensor by itself: the fully tangential and one-normal sectors leave the two-normal sector undetermined until normal evolution is supplied.

Write $K=h^{ab}K_{ab}$ and let $R$ denote the ambient scalar curvature. Contracting Gauss and Codazzi gives

$$R-2\epsilon R_{ab}n^a n^b ={}^{(h)}R-\epsilon\left(K^2-K_{ab}K^{ab}\right),$$ $$h_a{}^c n^dR_{cd}=D_aK-D_bK_a{}^b.$$

Equivalently, the scalar left-hand side is $-2\epsilon G_{ab}n^a n^b$, while the vector projection is unchanged when $R_{ab}$ is replaced by $G_{ab}$. These are geometric contraction identities. They become Hamiltonian and momentum constraints only after a gravitational field equation is imposed.

The explicit $\epsilon$ in the scalar identity is essential. For $\epsilon=-1$ it reads $R+2R_{nn}={}^{(h)}R+K^2-K_{ab}K^{ab}$; for $\epsilon=+1$, both normal-Ricci and quadratic-extrinsic terms reverse. The vector contraction has no additional $\epsilon$ because the displayed normal is already a contravariant unit vector. Changing its orientation reverses both sides.

Two-normal curvature

Extend the hypersurface locally to a non-null foliation and project the Lie derivative back into each leaf. Then the remaining nonzero projection is

$$h_a{}^c h_b{}^d n^e n^f R_{cedf} =\mathcal L_n K_{ab} +K_a{}^cK_{cb} +D_a a_b -\epsilon a_a a_b.$$

Here $\mathcal L_nK_{ab}$ means the tangentially projected Lie derivative. For a spacelike foliation, $\epsilon=-1$ and $a_a=D_a\ln N$, so the last two terms combine as

$$D_a a_b+a_a a_b=N^{-1}D_aD_bN,$$

recovering Gourgoulhon 2007, §3.4, Eqs. (3.38)–(3.43), printed pp. 47–48, PDF. Unlike Gauss and Codazzi, this formula depends on normal evolution and therefore on a local extension of the slice.

The acceleration terms cannot be dropped merely because the normal is unit. They vanish only for geodesic normal flow. In an adapted spacelike foliation a spatially varying lapse produces nonzero $a_a$, even when $K_{ab}=0$ and the ambient spacetime is flat. This is why the Rindler check below distinguishes the $D_a a_b-\epsilon a_a a_b$ combination from several plausible but incorrect sign choices.

Four discriminating checks

Flat hyperplane. In flat space with a constant normal, $K_{ab}=a_b=0$ and all three projections vanish.

Round sphere. For a radius-$r$ sphere in Euclidean space with outward normal, $\epsilon=+1$ and $K_{ab}=-h_{ab}/r$. Ambient curvature is zero, and Gauss returns sectional curvature $+1/r^2$.

Milne slice. A constant-$\tau$ hyperboloid in Minkowski space has $\epsilon=-1$, $K_{ij}=-h_{ij}/\tau$, and $a_i=0$. Gauss returns sectional curvature $-1/\tau^2$, while $\mathcal L_nK_{ij}+K_i{}^kK_{kj}=0$ checks the two-normal sign.

Rindler lapse. For $ds^2=-x^2d\eta^2+dx^2+d\mathbf y^2$ with $x\gt0$, the $\eta=$ constant slices have $K_{ij}=0$ and $a=d\ln x$. Since $D_xa_x=-x^{-2}$ and $-\epsilon a_xa_x=+x^{-2}$, the acceleration terms cancel, as flat ambient curvature requires.

These checks probe different parts of the package: the hyperplane tests the zero case, the sphere and Milne slice distinguish the two values of $\epsilon$, and Rindler isolates normal acceleration from extrinsic curvature. Passing only the hyperplane check cannot determine any relative sign.

The identities are local and necessary compatibility conditions. They do not by themselves prove that prescribed intrinsic and extrinsic data admit a global embedding, nor do they supply matter sources or evolution equations. Null hypersurfaces are excluded: their induced metric is degenerate, their normal cannot be normalized, and their geometry is not obtained by setting $\epsilon=0$.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. arXiv:gr-qc/0703035v1, lecture notes, 2007. arXiv record.
• Wendl, Chris. Differential Geometry I and II. Humboldt-Universität zu Berlin lecture notes, 2021–2022. Open PDF.