Curvature Analysis

Once velocity is represented on the cell state manifold, FlowMap can analyze how the vector field changes along its own trajectories. Curvature summarizes whether a local flow is approximately straight, turns within the manifold, or bends because the manifold itself is curved in gene expression space.

Let \(x(t)\) be a trajectory on the reconstructed expression manifold, with velocity and acceleration

\[v = \frac{dx}{dt}, \qquad a = \frac{dv}{dt}.\]

The magnitude of acceleration depends on the speed of the trajectory. To focus on bending rather than speed, FlowMap uses the speed-normalized quantity

\[\kappa = \frac{\lVert a \rVert}{\lVert v \rVert^2}.\]

This is the usual scaling for curvature: the same geometric curve should not look more curved merely because it is traversed faster.

Acceleration separates into a component along the flow, a tangent component that changes direction, and a normal component induced by the surface.

The component of \(a\) parallel to \(v\) changes speed. Curvature is controlled by the remaining components: a tangent steering term and a surface-normal term.

\[\kappa^2 = \kappa_{\mathrm{steer}}^2 + \kappa_{\mathrm{surface}}^2.\]

The steering term is the part of acceleration that lies in the tangent space but is orthogonal to the velocity. It changes the direction of the flow curve within the manifold.

Steering acceleration is tangent to the manifold and perpendicular to the velocity, so it turns the flow curve.

The surface term is the normal component of acceleration. It appears when the trajectory follows a curved manifold embedded in gene expression space. Even if the flow direction changes little within the tangent plane, the lifted trajectory can still bend in the ambient space.

Surface acceleration points normal to the manifold and measures how the expression surface bends along the trajectory.