Connection

A connection (or covariant derivative) tells you how to differentiate vector fields on a manifold; the fundamental structure needed to define curvature and the Riemannian Hessian.

Definition

Definition null (Levi-Civita Connection).

The Levi-Civita connection $\nabla$ is the unique affine connection on a Riemannian manifold $(M, g)$ that is:

Torsion-free: $\nabla_X Y - \nabla_Y X = [X, Y]$
Metric-compatible: $X\langle Y, Z\rangle = \langle \nabla_X Y, Z\rangle + \langle Y, \nabla_X Z\rangle$

In local coordinates it is encoded by the Christoffel symbols .

The `Connection` Trait

View in rustdoc →

pub trait Connection: Manifold {
  /// Covariant derivative of V along the curve with velocity X at p.
  fn covariant_derivative(
      &self,
      p: &Self::Point,
      x: &Self::Tangent,   // velocity direction
      v: &Self::Tangent,   // vector field to differentiate
  ) -> Self::Tangent;

  /// Christoffel symbols Γ^k_ij at p (optional override for efficiency).
  fn christoffel(
      &self,
      p: &Self::Point,
  ) -> [[[f64; Self::AMBIENT_DIM]; Self::AMBIENT_DIM]; Self::AMBIENT_DIM] {
      // default: computed from covariant_derivative via finite differences
      todo!()
  }
}

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

Euclidean Connection (Trivial Case)

On , the Levi-Civita connection is just the ordinary directional derivative; all Christoffel symbols vanish:

Example ?: Euclidean covariant derivative

For a vector field $V(p) = Ap$ (linear), $\nabla_X V = AX$, just matrix-vector multiplication.

use cartan::prelude::*;
use cartan::manifolds::Euclidean;
 
let r3 = Euclidean::<3>;
let p  = [1.0_f64, 0.0, 0.0];
let x  = [0.0_f64, 1.0, 0.0];  // differentiate in y-direction
let v  = [0.0_f64, 0.0, 1.0];  // constant vector field
 
// Covariant derivative of a constant field = 0.
let dv = r3.covariant_derivative(&p, &x, &v);
assert!(dv.iter().all(|&c| c.abs() < 1e-12));

Sphere Connection (Weingarten Map)

On , the Levi-Civita connection is the tangential projection of the ambient derivative:

The term $\langle DV \cdot X, p \rangle p$ is the Weingarten correction - it removes the normal (radial) component.

Riemannian Hessian

The connection enables second-order optimization via the Riemannian Hessian:

Definition null (Riemannian Hessian).

For , the Riemannian Hessian at $p$ is the symmetric operator defined by:

The Hessian-vector product (HVP) can be computed efficiently via a single Pearlmutter forward-over-reverse pass without materialising the full Hessian.

// HVP callback pattern used by cartan-optim.
let hvp = |u: &Tangent| manifold.covariant_derivative(&p, u, &grad_f);

Geodesics via Connection

The geodesic equation is precisely the condition that the velocity field is parallel along itself:

In coordinates this becomes

a second-order ODE whose solutions are the geodesics. cartan solves this analytically for each manifold rather than numerically integrating.