Exponential map on manifold

Read the equation

This relation defines exp_p(v) ≈ p + v - ½Γ(v,v) in the source section “Exponential map on manifold”. It specifies the displayed relationship between the named quantities or parameters. Its interpretation depends on the source’s stated variables and conditions; this record does not assign an unprovided physical or cognitive meaning to them.

Terms & definitions

The recovered passage does not supply a separate symbol table. Read the source context for the notation used in this version; familiar letters may carry different meanings in other builds.

How it works

1. Read the displayed expression together with the surrounding source statement; it is retained as an expression rather than silently completed into an equality.

2. Substitute the source-defined variables or recorded values and follow the displayed arithmetic and indexing exactly.

3. Use this version’s variable definitions and update order. The same symbol may be assigned a different meaning in another build.

Assumptions & limits

• This source expression was recovered from the pinned code revision. Its presence does not establish that the associated program was built, run, or scientifically validated.

Original context

 */
inline double ricci_scalar(double curvature) {
    return 2.0 * curvature;
}

/**
 * @brief Exponential map on manifold
 * exp_p(v) ≈ p + v - ½Γ(v,v)
 */
inline double exp_map(double p, double v, double christoffel) {
    return p + v - 0.5 * christoffel * v * v;
}

/**
 * @brief Parallel transport of vector along geodesic
 */
inline double parallel_transport(double v, double christoffel, double tangent, double dt) {
    return v - christoffel * v * tangent * dt;

exp_p(v) ≈ p + v - ½Γ(v,v)

Related builds