Gauss equation

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

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 $X,e_2,\dots ,e_n$,

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 ${\mathbb{R}}^n$.

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.