Page 144 - Curvature and Homology

126 III. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY

B. 1-parameter local groups of local transformations
1. Let P be a point on the differentiable manifold M and U a coordinate
neighborhood of P on which a vector field X # 0 is given. Denote the com-
ponents of X at P with respect to the natural basis in U by ti. There exists at P
a local coordinate system vl,---, vn such that the corresponding parametrized
curves with v1 as parameter have at each point Q the vector XQ as tangent vector.
If we put v1 = t, the equations
ui = ui(v2, .-. , vn, t), i = 1, *.. , n

defining the coordinate transformations at P are the equations of the 'integral
curves' (cf. I. D.8) when the vi, i = 2, ..., n are regarded as constants and t as
the parameter, that is, the coordinate functions ui, i = 1, ***, n are solutions of
the system of differential equations

with ('(0) = p, the point P corresponding to t = 0. More precisely, it is
possible to find a neighborhood U(Q) of Q and a positive number E(Q) for every
Q E U such that the system (*) has a solution for I t 1 < a(Q). Denoting this
solution by
ut(v2, , vn, t) = exp (tX)ui(v2, ... , vn, 0)
show that

provided both sides are defined. In this way, we see that the 'exp' map defines
a local 1-parameter group exp(tX) of (local) transformations.
2. Conversely, every 1-parameter local group of local transformations tp, may be
so defined. Indeed, for every P E M put

and consider the vector field X defined by the initial conditions

(or, rYp = (dP(t)/dt),,,). It follows that

3. The map exp(tX) is defined on a neighborhood U(Q) for I t I < c(Q) and
induces a map exp(tX), which is an isomorphism of Tp onto Tp(,,-the tangent