Page 102 - Curvature and Homology

84 11s. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY
on S in which distance plays no role but angle may be defined, that is
angle is invariant under a conformal change of coordinates. After
performing a birational transformation of the equation R(z, w) = 0 a
new algebraic equation is obtained. The Riemann surface S' of the
algebraic function thus obtained is homeomorphic to S. Let f : S -+ S'
denote the homeomorphism and (u, v), (u', v') the local coordinates
at P E S and P' = f(P) E S', respectively. The functions

are then analytic, that is f is a holomorphic homeomorphism. It follows
that
dut2 + dvIa = a2(dua + d$)
where a is an analytic function of u and v, that is the homeomorphism
is a conformal map of S onto Sf.
Conversely, functions whose Riemann surfaces are conformally
homeomorphic are birationally equivalent. Their Riemann surfaces
are then said to be equivalent.
A 2-dimensional Riemannian manifold and a Riemann surface are
both topological 2-manifolds. As differentiable manifolds however,
they differ in their differentiable structures-the former allowing
systems of local parameters related by functions with non-vanishing
Jacobian whereas in the latter case only those systems of local para-
meters which are conformally related are permissible. Clearly then,
they differ in their local geometries-the former being Riemannian
geometry whereas the latter is conformal geometry. To construct a
Riemann surface from a given 2-dimensional Riemannian manifold M
we need only restrict the systems of local coordinates so that in the
overlap of two coordinate neighborhoods the coordinates are related
by analytic functions defining a conformal transformation. That such a
covering of M exists follows from the possibility of introducing isothermal
parameters on M. The manifold is then said to possess a complex
(analytic) structure. We conclude that conformally homeomorphic
2-dimensional Riemannian manifolds defne equivalent Riemann surfaces.
The concept of a complex structure on an n(= 2m)-dimensional
topological manifold will be discussed in Chapter V.
Two n-dimensional Riemannian manifolds M and M' of class k are
said to be isometric if there is a differentiable homeomorphism f (of
class k) from M onto M' which maps one element of arc into the other.
It can be shown that a simply connected, complete Riemannian manifold
of constant curvature K is isometric with either Euclidean space (K = O),
hyperbolic space (K < 0), or spherical space (K > 0). Hence, the
universal covering manifold of a complete Riemannian manifold of