Page 125 - Curvature and Homology

3.7. CONFORMAL TRANSFORMATIONS
Proposition 3.7.1. For any vector jeld X

where 6 is the I-form on M dual to X.
Let U be a coordinate neighborhood with the local coordinates
ul, -.- un. The vector fields a/aul, .--, a/aun form a basis of the F-module
of vector fields in U where F is the algebra of differentiable functions
on U. Denoting the components of the metric tensor g by gij we have
g = gij dui @ duj. Applying the derivation B(X) to g we obtain

It follows that
+
ag 3 a tk a tk
(e(&)i~ = P j$ + +kj a +ik 9
and, since the right hand side is equal to Djfi + Diej we may write

Corollary. An infinitesimal conformal transformation X on an
n-dimensional Riemannian manifold satisfies the equation

Transvecting this equation with gij

Corollary. A necessary and sufficient condition that an infinitesimal
conformal transformation X be a motion is given by St = 0.
If the vector field X has constant divergence, that is, if Sf = const.,
the transformation is said to be homothetic.