Gauss–Codazzi equations

The Gauss–Codazzi equations relate the intrinsic curvature and connection of an embedded submanifold to its second fundamental form and the ambient curvature. The Gauss equation controls tangential curvature; the Codazzi equation controls the covariant derivative of the second fundamental form. The Gauss formula is the tangential-normal splitting used to derive these relations.

For a submanifold with induced Levi-Civita connection $\nabla$ and second fundamental form $A$, the ambient derivative decomposes asFor a parametrized surface this reads $X_{ij}={\Gamma }_{ij}^k{X}_k+h_{ij}N$.

For a Euclidean embedded submanifold,It expresses intrinsic curvature in terms of the second fundamental form.

For an embedded submanifold of a Riemannian manifold, use $R(v,w)y={\nabla }_v{\nabla }_{w}y-{\nabla }_w{\nabla }_{v}y-{\nabla }_{[v,w]}y$. The Gauss formula and tangential derivative of a normal field giveSubtract the expression with $v,w$ interchanged and the bracket derivative, whose normal part pairs to zero. This proves the displayed curvature identity. For flat Euclidean ambient space, its ambient curvature term vanishes and one obtains the ordinary Gauss equation. Declaring the four-slot convention $R(x,y,v,w)=⟨R(v,w)y,x⟩$ avoids sign ambiguities when permuting arguments.

For a two-plane $\sigma =\mathrm{span}(X,Y)$ in a Riemannian tangent space,

Ricci curvature is the trace of the Riemann curvature tensor in its first and third arguments. For a unit tangent vector $X$ and an orthonormal basis $X,e_2,\dots ,e_n$,

For an orthonormal triple and distinct $i,j,k$, tracing sectional curvature gives ${\mathrm{Ric}}_{ii}=K_{ij}+K_{ik}$. Solving the three equations gives the displayed formula. Equivalently $K_{ij}={\mathrm{Ric}}_{ii}+{\mathrm{Ric}}_{jj}-\mathrm{Scal}/2$. This pointwise assertion does not require the triple to diagonalize the Ricci curvature.

Scalar curvature is the complete trace of the Riemann curvature tensor:An $n$-manifold of constant sectional curvature $K$ has $\mathrm{Scal}=n(n-1)K$.

A Ricci-flat Riemannian manifold has identically zero Ricci tensor. Every flat Riemannian manifold is Ricci-flat, while the converse can fail in dimension at least four.

If a complete connected $n$-dimensional Riemannian manifold satisfies $\mathrm{Ric}\ge (n-1)kg$ for some $k>0$, then its diameter is at most $\pi /\sqrt{k}$. It is therefore compact and has finite fundamental group.

A complete connected Riemannian manifold with nonnegative Ricci curvature that contains a line splits isometrically as $N\times \mathbb{R}$.

A Riemannian manifold is flat when every sectional curvature is zero. A complete simply connected flat $n$-manifold is isometric to Euclidean space ${\mathbb{R}}^n$.

A flat torus is ${\mathbb{R}}^d/\Lambda$ with its quotient Euclidean metric, where $\Lambda$ is a full-rank Euclidean lattice. Its volume is the lattice covolume. The dual lattice indexes its complete Fourier series of Laplacian eigenfunctions.

With the nonnegative Laplace-Beltrami operator, a frequency $w$ in the dual lattice gives the eigenfunction $e^{2\pi i⟨w,x⟩}$ and eigenvalue $4{\pi }^2|w{|}^2$. Its multiplicity is the number of dual vectors with that norm. Fourier series prove completeness. In dimension two the shortest vector, shortest independent vector and covolume determine a reduced Gram matrix, proving spectral rigidity.

On a complete connected Riemannian manifold of nonpositive sectional curvature, the exponential map at every point is a covering map. If the manifold is simply connected, each exponential map is a diffeomorphism from a tangent space onto the manifold.

With outward normal $N(x)=x$, the unit sphere has $A(X,Y)=-⟨X,Y⟩N$. The Gauss equation therefore gives sectional curvature one on every tangent two-plane.

For a Euclidean embedded submanifold, the covariant derivative of the second fundamental form satisfies