the Lie-group exponential map
The exponential function e^x turns addition into multiplication: e^(s+t) = e^s e^t. The Lie-group exponential is the same idea wired into a curved group of symmetries. It takes an infinitesimal symmetry — an element of the Lie algebra, a 'velocity at the identity' — and integrates it for one unit of time to produce an actual group element. It is the master dictionary translating the linear algebra of g into the curved geometry of G.
Formally, for a Lie group G with Lie algebra g, the exponential map exp: g -> G sends X to gamma_X(1), where gamma_X is the unique one-parameter subgroup with gamma_X'(0) = X; equivalently exp(X) is the time-1 flow, started at the identity, of the left-invariant vector field determined by X. So exp(tX) = gamma_X(t) traces the whole one-parameter subgroup. For matrix groups this is literally the matrix exponential exp(A) = e^A = I + A + A^2/2! + A^3/3! + ... . Key properties: exp is smooth, exp(0) = e, its differential at 0 is the identity map on g (so exp is a diffeomorphism from a neighborhood of 0 in g onto a neighborhood of e in G — this is the source of 'canonical coordinates of the first kind'), and exp((s+t)X) = exp(sX) exp(tX). But beware: exp(X) exp(Y) is NOT exp(X+Y) unless X and Y commute; the correction is governed by the Baker-Campbell-Hausdorff formula, whose leading term is the bracket: exp(X)exp(Y) = exp(X + Y + (1/2)[X,Y] + ...).
The exponential map is how one-parameter subgroups, the adjoint representation, and the whole local theory get computed; it linearizes the group near the identity. Two honesty points specific to graduate work. First: exp need not be surjective onto a connected group — in SL(2, R) there are elements (certain ones with trace < -2) not of the form exp(X) — though for compact connected groups and for connected nilpotent groups it is onto. Second, and important: the Lie-group exp(X) and the Riemannian exp_p of Vol I are DIFFERENT constructions. They agree only for a bi-invariant metric (always available on compact groups), where one-parameter subgroups are geodesics through e; in general a left-invariant metric's geodesics are not one-parameter subgroups, so do not conflate the two exponentials.
Take X = [0, -theta; theta, 0] in so(2). Summing the matrix series exp(X) = I + X + X^2/2! + ... groups into the cosine and sine series and yields exp(X) = [cos theta, -sin theta; sin theta, cos theta], the rotation by theta. Here exp is surjective onto SO(2). By contrast, in SL(2, R) the element diag(-2, -1/2) (trace -5/2 < -2) lies in no image exp(X), so exp is not onto there.
Summing the matrix exponential of a rotation generator recovers cos and sin; in SL(2,R), exp is not surjective.
Do NOT conflate the Lie-group exp: g -> G with the Riemannian exp_p. They coincide only for a bi-invariant metric; a general left-invariant metric has geodesics that are not one-parameter subgroups. Also, exp need not be onto (e.g. SL(2,R)) or injective.