Trait RiemannMap 

pub trait RiemannMap<P: HKT4Unbound> {
    // Required methods
    fn curvature<A, B, C, D>(tensor: P::Type<A, B, C, D>, u: A, v: B, w: C) -> D;
    fn scatter<A, B, C, D>(
        interaction: P::Type<A, B, C, D>,
        in_1: A,
        in_2: B,
    ) -> (C, D);
}

The RiemannMap trait models high-arity geometric interactions, specifically the Riemann Curvature Tensor and Scattering Matrices.

Category Theory

This corresponds to a Multilinear Map in a Tensor Category. Specifically, the Curvature Tensor is a map $R: V \otimes V \otimes V \to V$.

Mathematical Definition

The Riemann Curvature Tensor $R$ is defined in terms of the covariant derivative $\nabla$: $$ R(u, v)w = \nabla_u \nabla_v w - \nabla_v \nabla_u w - \nabla_{[u, v]} w $$ It measures the non-commutativity of parallel transport around a loop defined by $u$ and $v$.

Use Cases

• General Relativity: Calculating gravity as spacetime curvature.
• Particle Physics: Scattering matrices (S-Matrix) taking 2 inputs and producing 2 outputs.
• Differential Geometry: Measuring the holonomy of a connection.

Required Methods

fn curvature<A, B, C, D>(tensor: P::Type<A, B, C, D>, u: A, v: B, w: C) -> D

The Curvature Operator: $R(u, v)w \to D$ Consumes two directions ($u, v$) and a vector ($w$) to measure curvature ($D$).

fn scatter<A, B, C, D>( interaction: P::Type<A, B, C, D>, in_1: A, in_2: B, ) -> (C, D)

The Scattering Matrix: $(A, B) \to (C, D)$ Models an interaction where two particles collide and produce two new states.

Dyn Compatibility

This trait is not dyn compatible.

In older versions of Rust, dyn compatibility was called "object safety", so this trait is not object safe.

Implementors