Page 124 - Curvature and Homology

106 . 111. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY

3.7. Conformal transformations
Let M be an n-dimensional Riemannian manifold and g the tensor field
of type (0,2) defining the Riemannian metric on M. Locally, the metric
is given by
d-sa = gif dug duf

where the gij are the components of g with respect to the natural frames
of a local coordinate system (ui). A metric g* on M is said to be con-
formally related tog if it is proportional to g, that is, if there is a function
p > 0 on M such that g* = p2g. By a conformal transformation of M
is meant a differentiable homeomorphism f of M onto itself with the
property that
f *(a'$) = p2 ds2

where f* is the induced map in the bundle of frames and p is a positive
function on M. Clearly, the set of conformal transformations of M forms
a group. In fact, it can be shown that it is a Lie transformation group.
Let G denote a connected Lie group of conformal transformations of M
and L its Lie algebra. To each element A EL is associated the l-para-
meter subgroup a, of G generated by A. The corresponding 1-parameter
group of transformations Rat on M induces a (right invariant) differenti-
able vector field A* on M. A* in turn defines an infinitesimal trans-
formation O(A*) of the tensor algebra over M corresponding to the action
on M of a,. From the action on the metric tensor g, it follows from (3.7.1)
that
8(A*)g = hg (3.7.2)
where A is a function depending on A*. On the other hand, a vector field
X on M which satisfies (3.7.2) is not necessarily complete (cf. 5 3.4).
However, X does generate a 1-parameter local group, and for this reason
X is called an infinitesimal conformal transformation of M. In our applica-
tions the manifold M will be compact and therefore the infinitesimal
conformal transformations will be complete. In any case, they form a
Lie algebra L with the usual bracket [X, YJ = O(X)Y.
If the scalar A vanishes, that is, if O(X)g = 0, the metric tensor g is
invariant under the action of O(X). The vector field X is then said to
define an injinitesimal motion. The infinitesimal motions define a sub-
algebra of the Lie algebra L. For, O([X, YJ)g = O(X)O(Y)g - O(Y)O(X)g
= 0. Moreover, it can be shown that the group of all the isometries of
M onto itself is a Lie group (with respect to the natural topology).
If 6 is the 1-form on M dual to X we shall occasionally write O(6)
for O(X).