The map exp_p from the tangent space T_pM at a point p of a (typically Riemannian) manifold M to M that sends a tangent vector v to the endpoint γ_v(1) of the geodesic γ_v with initial velocity v, defined on the maximal domain where those geodesics exist.

Principle

Geodesic flow linearizes infinitesimal directions: the exponential map translates straight (geodesic) motion in the linear tangent space into motion on the manifold, providing a local diffeomorphism from a neighborhood of 0 in T_pM onto a normal neighborhood of p.

Demonstration

On Euclidean space R^n the exponential map at any point is the affine identification exp_p(v)=p+v (global diffeomorphism); on the sphere S^n exp_p maps a radial tangent vector to the point obtained by following the great circle geodesic for unit time, with conjugate points limiting global injectivity.

Misapplication

Assuming exp_p is globally defined and injective for arbitrary manifolds; in general it is only locally defined/diffeomorphic near 0 and can fail to be injective because of conjugate points or cut loci.

Consequence

The exponential map yields normal coordinates, relates Riemannian distance to lengths of tangent vectors, and is central to many constructions (geodesic completeness, comparison theorems, defining the Riemannian logarithm locally).

Reversal

Locally invertible portions of exp_p admit a local inverse log_p mapping nearby manifold points back to tangent vectors; globally such an inverse may not exist, and the topology of M complicates inversion.

Boundary

Requires a connection (in Riemannian geometry the Levi-Civita connection) to define geodesics; in absence of a chosen affine connection or on incomplete manifolds the domain of the exponential map is restricted or undefined.

Semantic Tension

Frequently conflated with the matrix or Lie-group exponential which maps algebra elements to group elements; the manifold exponential is a geometric geodesic construction that coincides with the Lie exponential only in the special context of Lie groups with a bi-invariant connection.

Synthesis

The exponential map converts tangent vectors into points reached by geodesics and provides the principal link between the linear tangent model and the nonlinear manifold, enabling local coordinate systems, comparison geometry, and the (local) logarithm map.