Page 19 - Curvature and Homology

CHAPTER I

RIEMANNIAN MANIFOLDS

In seeking to generalize the well-known theorem of Riemann on the
existence of holomorphic integrals over a Riemann surface, W. V. D.
Hodge 1391 considers an n-dimensional Riemannian manifold as the
space over which a certain class of integrals is defined. Now, a Riemannian
manifbld of two dimensions is not a Riemann surface, for the geometry
of the former is Riemannian geometry whereas that of a Riemann surface
is conformal geometry. However, in a certain sense a 2-dimensional
Riemannian manifold may be thought of as a Riemann surface. More-
over, conformally homeomorphic Riemannian manifolds of two dimen-
sions define equivalent Riemann surfaces. Conversely, a Riemann
surface determines an infinite set of conformally homeomorphic 2-dimen-
sional Riemannian manifolds. Since the underlying structure of a
Riemannian manifold is a differentiable structure, we discuss in this
chapter the concept of a differentiable manifold, and then construct
over the manifold the integrals, tensor fields and differential forms
which are basically the objects of study in the remainder of this book.

1 .l. Differentiable manifolds

The differential calculus is the main tool used in the study of the
geometrical properties of curves and surfaces in ordinary Euclidean
space E9. The concept of a curve or surface is not a simple one, so that
in many treatises on differential geometry a rigorous definition is lacking.
The discussions on surfaces are further complicated since one is interested
in those properties which remain invariant under the group of motions
in @. This group is itself a 6-dimensional manifold. The purpose of
this section is to develop the fundamental concepts of differentiable
manifolds necessary for a rigorous treatment of differential geometry.
Given a topological space, one can decide whether a given function
1