The derivative of the exponential map

We give a quick analytic derivation of the formula for the derivative of the exponential of a vector field on a manifold. Our object is to give a quick proof of a result usually obtained only for analytic manifolds using combinatorial manipulation with power series. Let Xf be a C()-manifold. Put VEC(,#) for the space of C*)-vector fields on X#. Given X E VEC(#) and p E X we write exp(tX)p for the point on the manifold corresponding to the flow of X at time t that passes through p at time t = O. Thus one has (d/dt)f o exp(tX)It= = f' o X everywhere on X, where f' is the derivative of the mapping f: # -:+ R. If exp(tX) is globally defined then (d/dt)f o exp(tX) = P o X o exp(tX) = f o T(exp(tX)) o X. Here we use the notation T(S): T(Xf) -T(df) for the functorial map of the tangent bundle corresponding to a C(9)-map S: X# X. Let S be an automorphism of Xf. Given V E VEC(4t) we get a new vector field (AdS)Vd T(S)oVoS1. (1) Proposition. Suppose X, V E VEc(A) and that exp X is globally defined. Then, for any test function f e C(?) () we have (d/dt)f o (exp(X + tV)) o exp(-X)J I = f' of Ad(expsX) Vds. Remark. X and V are fixed in the statement. Thus, for each p E X, s F-4 Ad(expsX) V(p) gives a continuous map [0, 1] -+ Tp(A#). f' is a linear functional on the finite-dimensional vector space Tp(.). Thus one has an Received by the editors June 6, 1990 and, in revised form, November 21, 1990. 1991 Mathematics Subject Classification. Primary 58A30, 22E30.