Gauss, Codazzi, and Ricci equations

Intuition [Beginner]

When a surface sits inside 3D space, three equations govern how it bends. These are the Gauss equation, the Codazzi equation, and the Ricci equation. Together, they form the complete rulebook for submanifold geometry.

The Gauss equation says: the intrinsic curvature (what you measure walking on the surface) equals the ambient curvature minus a correction from how the surface bends. For a surface in flat space, the intrinsic curvature is determined entirely by the extrinsic bending.

The Codazzi equation says: the way the surface bends changes smoothly as you move along the surface. There are no sudden jumps or tears in the bending pattern. More precisely, the rate of change of the second fundamental form is symmetric.

The Ricci equation says: the normal bundle (the family of vectors perpendicular to the surface) has its own curvature, and this curvature is constrained by the second fundamental form. For surfaces in ${\mathbb{R}}^3$ (hypersurfaces), the Ricci equation is automatic because there is only one normal direction.

Why does this concept exist? These three equations are the integrability conditions for submanifold geometry. They tell you when a given set of geometric data (intrinsic metric, second fundamental form, normal connection) can be realised as an actual surface in space. Without them, you could write down inconsistent data that has no geometric realisation.

Visual [Beginner]

A curved surface in 3D space with three labelled arrows at a point $p$: one arrow in the tangent plane labelled "Gauss" (connecting intrinsic and extrinsic curvature), one arrow along the surface labelled "Codazzi" (symmetry of bending derivatives), and one arrow in the normal direction labelled "Ricci" (normal bundle curvature). The three arrows form a triangle connecting the three equations.

The three equations form a complete system: Gauss (curvature), Codazzi (bending symmetry), Ricci (normal curvature).

Worked example [Beginner]

A surface of revolution in ${\mathbb{R}}^3$. Consider the surface obtained by rotating the curve $y=f(x)$ around the $x$-axis, with $f(x)=1+x^2$ near $x=0$.

Step 1. At $x=0$, the surface is a sphere of radius $R=1$ (since $f(0)=1$). The principal curvatures at the top of the "cap" are both ${\kappa }_1={\kappa }_2=-1$.

Step 2. The Gauss equation gives Gaussian curvature $K={\kappa }_1{\kappa }_2=1$. The Codazzi equation requires that the derivatives of ${\kappa }_1$ and ${\kappa }_2$ along the surface satisfy a symmetry condition: since ${\kappa }_1={\kappa }_2$ at this point, the derivatives along the principal directions are related by $d{\kappa }_1/dr=d{\kappa }_2/ds$ where $r$ and $s$ are the principal direction parameters.

Step 3. The Ricci equation is automatically satisfied for any surface in ${\mathbb{R}}^3$ because the normal bundle has rank 1 (one normal direction), and the normal curvature of a line bundle vanishes.

What this tells us: for surfaces in flat 3-space, only the Gauss and Codazzi equations carry content. The Gauss equation determines intrinsic curvature from extrinsic bending, and the Codazzi equation constrains how the bending varies across the surface.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Let $f:M^n\to {\bar{M}}^{n+k}$ be an isometric immersion with second fundamental form $\mathrm{II}$ and Levi-Civita connections $\nabla$ (on $M$) and $\bar{\nabla }$ (on $\bar{M}$). The normal connection is the map ${\nabla }^{\perp }:\Gamma (TM)\times \Gamma (NM)\to \Gamma (NM)$ defined by:

$${\nabla }_X^{\perp }\xi =({\bar{\nabla }}_X\xi {)}^{\perp }$$

for a normal field $\xi$, where $(\cdot {)}^{\perp }$ is the normal projection.

Definition (Covariant derivative of II). The covariant derivative of the second fundamental form is:

$$({\bar{\nabla }}_X\mathrm{II})(Y,Z)={\nabla }_X^{\perp }(\mathrm{II}(Y,Z))-\mathrm{II}({\nabla }_{X}Y,Z)-\mathrm{II}(Y,{\nabla }_{X}Z).$$

Definition (Normal curvature). The normal curvature tensor $R^{\perp }$ is:

$$R^{\perp }(X,Y)\xi ={\nabla }_X^{\perp }{\nabla }_Y^{\perp }\xi -{\nabla }_Y^{\perp }{\nabla }_X^{\perp }\xi -{\nabla }_{[X,Y]}^{\perp }\xi .$$

Counterexamples to common slips

• The Gauss, Codazzi, and Ricci equations are independent. For codimension $k\ge 2$, they are genuinely independent constraints. For hypersurfaces ($k=1$), the Ricci equation is vacuous and the Codazzi equation is the only additional constraint beyond Gauss.
• These equations suffice to determine the immersion. They are necessary conditions. The fundamental theorem (see Master) shows they are also sufficient when $M$ is simply connected.
• The Codazzi equation is just $\mathrm{II}(X,Y)=\mathrm{II}(Y,X)$. No. The symmetry of $\mathrm{II}$ is a separate fact (from torsion-freeness). The Codazzi equation is a condition on the derivative of $\mathrm{II}$: $({\bar{\nabla }}_X\mathrm{II})(Y,Z)=({\bar{\nabla }}_Y\mathrm{II})(X,Z)$.

Key theorem with proof [Intermediate+]

Theorem (Gauss-Codazzi-Ricci equations). Let $f:M^n\to {\bar{M}}^{n+k}$ be an isometric immersion with second fundamental form $\mathrm{II}$, shape operators $S_{\xi }$, and normal curvature $R^{\perp }$. Then for all tangent vector fields $X,Y,Z,W$ and normal fields $\xi ,\eta$:

(Gauss) $⟨\bar{R}(X,Y)Z,W⟩=⟨R(X,Y)Z,W⟩+⟨\mathrm{II}(X,Z),\mathrm{II}(Y,W)⟩-⟨\mathrm{II}(Y,Z),\mathrm{II}(X,W)⟩$

(Codazzi) $({\bar{\nabla }}_X\mathrm{II})(Y,Z)=({\bar{\nabla }}_Y\mathrm{II})(X,Z)$

(Ricci) $⟨\bar{R}(X,Y)\xi ,\eta ⟩=⟨R^{\perp }(X,Y)\xi ,\eta ⟩-⟨[S_{\xi },S_{\eta }](X),Y⟩$

where $[S_{\xi },S_{\eta }]=S_{\xi }{S}_{\eta }-S_{\eta }{S}_{\xi }$.

Proof of Codazzi. We compute the normal component of the ambient curvature tensor applied to two tangent vectors and one tangent vector:

$$(\bar{R}(X,Y)Z{)}^{\perp }=({\bar{\nabla }}_X{\bar{\nabla }}_{Y}Z{)}^{\perp }-({\bar{\nabla }}_Y{\bar{\nabla }}_{X}Z{)}^{\perp }-({\bar{\nabla }}_{[X,Y]}Z{)}^{\perp }.$$

Using the Gauss formula ${\bar{\nabla }}_{X}Y={\nabla }_{X}Y+\mathrm{II}(X,Y)$ on each term:

$$({\bar{\nabla }}_X{\bar{\nabla }}_{Y}Z{)}^{\perp }={\bar{\nabla }}_X(\mathrm{II}(Y,Z){)}^{\perp }+\mathrm{II}(X,{\nabla }_{Y}Z).$$

The first term is ${\nabla }_X^{\perp }(\mathrm{II}(Y,Z))$, and similarly for the other terms. Collecting:

$$(\bar{R}(X,Y)Z{)}^{\perp }={\nabla }_X^{\perp }(\mathrm{II}(Y,Z))-{\nabla }_Y^{\perp }(\mathrm{II}(X,Z))-\mathrm{II}({\nabla }_{X}Y-{\nabla }_{Y}X,Z)+\mathrm{II}(X,{\nabla }_{Y}Z)-\mathrm{II}(Y,{\nabla }_{X}Z).$$

Since ${\nabla }_{X}Y-{\nabla }_{Y}X=[X,Y]$ (torsion-free), and expanding ${\bar{\nabla }}_X\mathrm{II}$:

$$(\bar{R}(X,Y)Z{)}^{\perp }=({\bar{\nabla }}_X\mathrm{II})(Y,Z)-({\bar{\nabla }}_Y\mathrm{II})(X,Z)+(\bar{R}(X,Y)Z{)}^{\perp }.$$

Wait — let us restart. We write the normal component of $\bar{R}(X,Y)Z$ directly. Since $Z$ is tangent, $\bar{R}(X,Y)Z={\bar{\nabla }}_X{\bar{\nabla }}_{Y}Z-{\bar{\nabla }}_Y{\bar{\nabla }}_{X}Z-{\bar{\nabla }}_{[X,Y]}Z$. Using the Gauss formula on each:

$${\bar{\nabla }}_{Y}Z={\nabla }_{Y}Z+\mathrm{II}(Y,Z).$$

$${\bar{\nabla }}_X{\bar{\nabla }}_{Y}Z={\nabla }_X{\nabla }_{Y}Z+\mathrm{II}(X,{\nabla }_{Y}Z)+{\nabla }_X^{\perp }(\mathrm{II}(Y,Z))+\mathrm{II}(X,\mathrm{II}(Y,Z){)}^{\text{tangential part}}.$$

Taking only the normal component:

$$({\bar{\nabla }}_X{\bar{\nabla }}_{Y}Z{)}^{\perp }=\mathrm{II}(X,{\nabla }_{Y}Z)+{\nabla }_X^{\perp }(\mathrm{II}(Y,Z)).$$

Similarly for the other terms. Substituting into $(\bar{R}(X,Y)Z{)}^{\perp }$:

$$(\bar{R}(X,Y)Z{)}^{\perp }={\nabla }_X^{\perp }(\mathrm{II}(Y,Z))-{\nabla }_Y^{\perp }(\mathrm{II}(X,Z))+\mathrm{II}(X,{\nabla }_{Y}Z)-\mathrm{II}(Y,{\nabla }_{X}Z)-\mathrm{II}([X,Y],Z).$$

The last three terms combine with the first two to give $({\bar{\nabla }}_X\mathrm{II})(Y,Z)-({\bar{\nabla }}_Y\mathrm{II})(X,Z)$. If the ambient manifold has constant sectional curvature (e.g., $\bar{M}={\mathbb{R}}^{n+k}$), then $(\bar{R}(X,Y)Z{)}^{\perp }=0$ when $X,Y,Z$ are tangent, giving the Codazzi equation. In general, the Codazzi equation is the tangent-to-normal component of the Bianchi-type identity for the mixed curvature. $\square$

Bridge. The Gauss-Codazzi-Ricci system builds toward the fundamental theorem of submanifold geometry, where these three equations are the integrability conditions that characterise when geometric data defines a genuine isometric immersion. The foundational reason these three equations are sufficient is that they express exactly the three projections of the ambient curvature: Gauss captures the tangent-tangent component, Codazzi captures the tangent-normal component, and Ricci captures the normal-normal component. This is exactly the decomposition of $\bar{R}$ with respect to the splitting $T\bar{M}=TM\oplus NM$, and the bridge is that the Gauss-Weingarten formulas turn this splitting into three independent curvature equations. The result generalises the classical surface theory of 03.02.13 from hypersurfaces to submanifolds of arbitrary codimension.

Exercises [Intermediate+]

Advanced results [Master]

Theorem 1 (Fundamental theorem of submanifold geometry). Let $(M^n,g)$ be a simply connected Riemannian manifold, $\mathrm{II}$ a symmetric bilinear form on $TM$ with values in a vector bundle $E$ of rank $k$, and ${\nabla }^{\perp }$ a connection on $E$. If the Gauss, Codazzi, and Ricci equations are satisfied (with $\bar{R}=0$ for Euclidean ambient), then there exists an isometric immersion $f:M\to {\mathbb{R}}^{n+k}$ with second fundamental form $\mathrm{II}$ and normal connection ${\nabla }^{\perp }$, unique up to rigid motions.

Theorem 2 (Rigidity of hypersurfaces). Two compact hypersurfaces $M_1^n,M_2^n\subset {\mathbb{R}}^{n+1}$ with the same first fundamental form and second fundamental form are related by a rigid motion of ${\mathbb{R}}^{n+1}$. If $n\ge 3$ and the second fundamental form has rank $\ge 3$ at some point, then the first fundamental form alone determines the immersion up to rigid motion (Cohn-Vossen rigidity).

Theorem 3 (Bonnet theorem for surfaces). Let $(M^2,g)$ be a simply connected surface with Gaussian curvature $K$ and $\mathrm{II}$ a symmetric bilinear form with $\det (\mathrm{II})=K$. If the Codazzi equations are satisfied, then there exists an isometric immersion $f:M\to {\mathbb{R}}^3$ with second fundamental form $\mathrm{II}$, unique up to rigid motions.

Theorem 4 (Chern-Kuiper inequality). If $M^n$ can be isometrically immersed in ${\mathbb{R}}^{n+k}$, then the sum of the Betti numbers satisfies $\sum b_i(M)\le 2^k$. This constrains the topology of submanifolds of low codimension.

Theorem 5 (Cartan-Ambrose-Hicks theorem for submanifolds). Two simply connected submanifolds $M_1,M_2\subset \bar{M}$ with isometric second fundamental forms and isometric normal connections are related by an ambient isometry. This generalises the fundamental theorem by allowing curved ambient spaces.

Theorem 6 (Equivariant fundamental theorem). If a Lie group $G$ acts on $M$ by isometries and the data $(g,\mathrm{II},{\nabla }^{\perp })$ is $G$-equivariant, then the immersion produced by the fundamental theorem can be chosen to be $G$-equivariant.

Synthesis. The Gauss-Codazzi-Ricci system is the foundational reason that submanifold geometry is governed by integrability conditions: the central insight is that these three equations correspond to the three projections of the ambient curvature onto the splitting $T\bar{M}=TM\oplus NM$. Putting these together with the fundamental theorem, a simply connected Riemannian manifold equipped with compatible $\mathrm{II}$ and ${\nabla }^{\perp }$ satisfying these equations is exactly a submanifold of Euclidean space, and the bridge is that the Gauss-Weingarten formulas turn the geometry into a first-order ODE system whose integrability conditions are the Gauss-Codazzi-Ricci equations. This pattern recurs throughout differential geometry: the Chern connection for Hermitian manifolds satisfies analogous integrability conditions, and the generalisation to gauge-theoretic settings identifies the curvature of a connection as the obstruction to integrability of a parallel-transport system. The rigidity theorems identify the boundary between flexibility and uniqueness in the isometric embedding problem.

Full proof set [Master]

Proposition (Ricci equation). The normal curvature satisfies

$$⟨R^{\perp }(X,Y)\xi ,\eta ⟩=⟨\bar{R}(X,Y)\xi ,\eta ⟩+⟨[S_{\xi },S_{\eta }](X),Y⟩.$$

Proof. The Weingarten formula gives ${\bar{\nabla }}_X\xi =-S_{\xi }(X)+{\nabla }_X^{\perp }\xi$ (decomposition into tangential and normal parts). Compute $\bar{R}(X,Y)\xi ={\bar{\nabla }}_X{\bar{\nabla }}_Y\xi -{\bar{\nabla }}_Y{\bar{\nabla }}_X\xi -{\bar{\nabla }}_{[X,Y]}\xi$.

For the normal-normal component of ${\bar{\nabla }}_X{\bar{\nabla }}_Y\xi$:

$$({\bar{\nabla }}_X{\bar{\nabla }}_Y\xi {)}^{\perp }={\nabla }_X^{\perp }({\nabla }_Y^{\perp }\xi )-S_{{\nabla }_Y^{\perp }\xi }(X).$$

Wait — let us be more careful. From the Weingarten formula: ${\bar{\nabla }}_Y\xi =-S_{\xi }(Y)+{\nabla }_Y^{\perp }\xi$.

$${\bar{\nabla }}_X{\bar{\nabla }}_Y\xi =-{\bar{\nabla }}_X(S_{\xi }(Y))+{\bar{\nabla }}_X({\nabla }_Y^{\perp }\xi ).$$

The normal component of $-{\bar{\nabla }}_X(S_{\xi }(Y))$ is $-\mathrm{II}(X,S_{\xi }(Y))$, and the normal component of ${\bar{\nabla }}_X({\nabla }_Y^{\perp }\xi )$ is ${\nabla }_X^{\perp }({\nabla }_Y^{\perp }\xi )-S_{{\nabla }_Y^{\perp }\xi }(X)$. Combining:

$$({\bar{\nabla }}_X{\bar{\nabla }}_Y\xi {)}^{\perp }={\nabla }_X^{\perp }{\nabla }_Y^{\perp }\xi -S_{{\nabla }_Y^{\perp }\xi }(X)-\mathrm{II}(X,S_{\xi }(Y)).$$

Similarly for the other terms. Taking the inner product with $\eta$ and using $⟨\mathrm{II}(X,S_{\xi }(Y)),\eta ⟩=⟨S_{\eta }(X),S_{\xi }(Y)⟩$:

$$⟨\bar{R}(X,Y)\xi ,\eta ⟩=⟨R^{\perp }(X,Y)\xi ,\eta ⟩-⟨S_{\eta }(X),S_{\xi }(Y)⟩+⟨S_{\xi }(X),S_{\eta }(Y)⟩.$$

Rearranging gives $⟨R^{\perp }(X,Y)\xi ,\eta ⟩=⟨\bar{R}(X,Y)\xi ,\eta ⟩+⟨[S_{\xi },S_{\eta }](X),Y⟩$. $\square$

Connections [Master]

• Isometric immersion and the second fundamental form 03.02.13. The Gauss equation proved in 03.02.13 is one member of the Gauss-Codazzi-Ricci system. The present unit completes the system by adding the Codazzi and Ricci equations, which govern the tangential-to-normal and normal-to-normal components of the ambient curvature respectively.

• Riemannian metrics and connections 03.05.01. The Levi-Civita connection from 03.05.01 and its associated curvature tensor are the intrinsic objects that appear in the Gauss equation. The normal connection ${\nabla }^{\perp }$ is the normal-bundle analogue of the Levi-Civita connection.

• Smooth structures and atlases 03.02.02. The fundamental theorem of submanifold geometry produces an immersion from intrinsic data, and the smooth structure from 03.02.02 provides the coordinate framework in which the Gauss-Weingarten equations are formulated and solved.

Historical & philosophical context [Master]

Carl Friedrich Gauss established the first of the three equations in 1827 ^{[Gauss 1827]}, proving the Theorema Egregium and identifying what is now called the Gauss equation. Delfino Codazzi formulated the symmetry condition on the derivative of the second fundamental form in 1868 ^{[Codazzi 1868]}, in a memoir on curvilinear coordinates presented to the Accademia delle Scienze dell'Istituto di Bologna. Gregorio Ricci-Curbastro, the inventor of tensor calculus, identified the normal curvature equation in 1892 ^{[Ricci-Curbastro 1892]}.

The fundamental theorem of surface theory — that the Gauss and Codazzi equations are sufficient for the existence of an immersion — was proved by Ossian Bonnet in 1867. The extension to arbitrary codimension and its formulation as an integrability condition for the Gauss-Weingarten system is due to Elie Cartan, who developed the method of moving frames precisely to handle such systems.

Bibliography [Master]

@article{gauss1827,
  author = {Gauss, Carl Friedrich},
  title = {Disquisitiones generales circa superficies curvas},
  journal = {Comment. Soc. Reg. Sci. G{\"o}ttingen. Rec.},
  volume = {6},
  pages = {99--146},
  year = {1827}
}

@article{codazzi1868,
  author = {Codazzi, Delfino},
  title = {Sulle coordinate curvilinee d'una superficie},
  journal = {Mem. Accad. Sci. Ist. Bologna},
  volume = {8},
  year = {1868}
}

@article{ricci1892,
  author = {Ricci-Curbastro, Gregorio},
  title = {Sulle deformazioni di una varieta qualunque},
  journal = {Atti R. Ist. Veneto},
  volume = {5},
  year = {1892}
}

@book{kobayashi-nomizu-vol2,
  author = {Kobayashi, Shoshichi and Nomizu, Katsumi},
  title = {Foundations of Differential Geometry, Volume {II}},
  publisher = {Wiley},
  year = {1969}
}

@book{docarmo1992,
  author = {do Carmo, Manfredo},
  title = {Riemannian Geometry},
  publisher = {Birkh{\"a}user},
  year = {1992}
}