ExponentialMap

Exponential map is a fundamental construction in differential geometry that relates the tangent space at a point to the ambient manifold. In a Riemannian manifold (M, g), the exponential map at p ∈ M, denoted exp_p, sends a tangent vector v ∈ T_pM to the point on M reached by following the unique geodesic starting at p with initial velocity v, after unit time. Concretely, if γ_v is the geodesic with γ_v(0) = p and γ_v'(0) = v, then exp_p(v) = γ_v(1). The map is defined on a neighborhood of the origin in T_pM, and, when the manifold is complete, on all of T_pM.

For Lie groups, the exponential map connects the Lie algebra 𝔤 to the group G. Given G, exp: 𝔤 → G is defined by exp(X) = γ_X(1), where γ_X is the unique one-parameter subgroup with γ_X'(0) = X. In matrix Lie groups, this agrees with the ordinary matrix exponential exp(X) = ∑ X^k / k!.

Key properties include that exp_p is a local diffeomorphism near 0, so normal coordinates can be defined via its inverse where defined; the differential at 0 is the identity on T_pM. Globally, behavior varies: for connected compact Lie groups, exp is surjective; for many noncompact groups it need not be.

Examples illustrate the concept. In Euclidean space with the standard metric, exp_p(v) = p + v. For SO(n), the exponential map sends skew-symmetric matrices to rotation matrices. In Riemannian geometry, the exponential map provides a tool for comparing distances and constructing normal coordinate systems around p.