Curvature

Curvature measures how much a manifold deviates from flat Euclidean space. cartan exposes three scalar/tensor quantities: sectional, Ricci, and scalar.

Riemann Curvature Tensor

Definition null (Riemann Curvature Tensor).

The Riemann curvature tensor $R$ is the $(1,3)$-tensor:

It measures the failure of covariant differentiation to commute. On a flat manifold $R \equiv 0$.

Sectional Curvature

Definition null (Sectional Curvature).

For a 2-plane , the sectional curvature is:

This is the Gaussian curvature of the geodesic surface swept out by $\sigma$.

Example ?: Sectional curvature on S²

For any 2-plane at any point, $K \equiv 1$ on the unit sphere. This matches the Gaussian curvature formula $K = 1/R^2$ for a sphere of radius $R = 1$.

use cartan::prelude::*;
use cartan::manifolds::Sphere;
 
let s2  = Sphere::<3>;
let mut rng = rand::rng();
let p   = s2.random_point(&mut rng);
let u   = s2.random_tangent(&p, &mut rng);
let v   = s2.random_tangent(&p, &mut rng);
 
let k = s2.sectional(&p, &u, &v);
assert!((k; 1.0).abs() < 1e-10);   // K ≡ 1 everywhere on S²

Known Values

Manifold	Sectional $K$	Ricci	Scalar $s$
	$\equiv 1$	$(N-2)\langle u,v\rangle$	$(N-1)(N-2)$
	$\equiv 0$	$0$	$0$
$SO(N)$
	$K \leq 0$ (NPC)	-	-
	$K \in [0, 2]$	-	-

The `Curvature` Trait

View in rustdoc →

pub trait Curvature: Connection {
  /// Sectional curvature of the 2-plane spanned by u, v at p.
  fn sectional(
      &self,
      p: &Self::Point,
      u: &Self::Tangent,
      v: &Self::Tangent,
  ) -> f64;

  /// Ricci tensor: Ric(u, v) = tr(w ↦ R(w, u)v).
  fn ricci_tensor(
      &self,
      p: &Self::Point,
      u: &Self::Tangent,
      v: &Self::Tangent,
  ) -> f64;

  /// Scalar curvature: s = tr(Ric) = sum of Ricci eigenvalues.
  fn scalar_curvature(&self, p: &Self::Point) -> f64;
}

Implemented by: Sphere<N>, Euclidean<N>, SO<N>

Algebraic Bianchi Identity

Theorem null (Algebraic Bianchi Identity).

The Riemann tensor satisfies:

Sphere Riemann Tensor

On , the Riemann tensor takes the especially clean form:

This is the tensor of a space form with constant sectional curvature $K = 1$. The same formula holds for any sphere of radius $r$ with $K = 1/r^2$.

Geometric Interpretation

Positive sectional curvature ($K > 0$): geodesics initially parallel converge (like lines of longitude on a sphere). Negative sectional curvature ($K < 0$): geodesics diverge exponentially (hyperbolic space). Zero ($K = 0$): Euclidean behavior, geodesics stay parallel.

This directly affects optimization: on positively curved manifolds the Riemannian gradient descent step is more conservative near the cut locus; on negatively curved (NPC/CAT(0)) manifolds like SPD, the Fréchet mean is unique and gradient descent converges globally.