Page 48 - Curvature and Homology

30 I. RIEMANNIAN MANIFOLDS

space T, at x E B onto the tangent space T,.,. This, in turn gives rise
to an isomorphism of T:., onto T,*. On the other hand, the projection
map w induces a map T* of T: (the space of covectors at P E M). An
affine connection on M may then be described as follows:
(i) T,* is the direct sum of W,* and w*(T,*) where Wz is a linear
subspace at x E B and n(x) = P;
by
(ii) For every a E GL(n, R) and x E B, is the image of c.,
the induced map on the space of covectors.
In other words, an aflne connection on M is a choice of a subspace W*,
in T,* at each point x of B subject to the conditions (2') and (it]. Note that
the dimension of W,* is n2. Hence, it can be defined by prescribing n2
linearly independent differential forms which together with the 85
span Tz.

1.9. Riemannian geometry
Unless otherwise indicated, we shall assume in the sequel that we are
given a differentiable manifold M of dimension n and class 00.
A Riemannian metric on M is a tensor field g of type (0,2) on M subject
to the conditions:
(i) g is a symmetric tensor field, and
(ii) g is positive definite.
This tensor field is called the fundamental tensor field. When a Riemannian
metric is given on M the manifold is called a Riemannian manifold.
Geometry based upon a Riemannian metric is called Riemannian
geometry. A Riemannian metric gives rise to an inner (scalar) product
on each tangent space Tp at P E M: the scalar product of the contra-
variant vector fields X = 64(a/hc) and Y = qi(a/aui) at the point P
is defined to be the scalar

The positive square root of X X is called the length of the vector X.
Since the Riemannian metric is a tensor field, the quadratic differential
form
d.@ = gjk duj duk (1 .9.2)
(where we have written duj duk in place of duj @ duk for convenience)