Page 52 - Curvature and Homology

34 I. RIEMANNIAN MANIFOLDS

be geometrically described by saying that the tangent spaces along C(t)
are developed into An and compared by means of the development.
An eleme~lt of fi over P E M is a set of n mutually perpendicular unit
vectors XI, ..., Xn in the tangent space at P. The frames along C are
developed into affine space An and, as before, the images are denoted
by a prime, so that P -+ P' and X, -+ Xi (i = 1,2, ..., n). In this way,
a scalar product may be defined in An by identifying An with one of its
tangent spaces and putting

Since the Levi Civita parallelism is an isometric linear map f, between
tangent spaces, the scalar product defined in An has an invariant meaning;
for, f,X f, Y = X Y.
Since the vectors of a frame are contravariant vectors, they determine
a set of n linearly independent vectors in the space of covectors at the
same point P, and since this latter space may be identified with A~(T;)
a frame at P defines a set of independent 1-forms 8, at that point. We
make a change in our notation at this stage: Since we deal with a
development of the tangent spaces along C into the vector space An
we shall denote by P,{ee,, ..., en) a typical frame in B over P so that the
image frame P1,{e;, ..., eA)(P -+ P') in An is a 'fixed' basis for the frames
in An. Now, consider the vectorial 1-form X?,, 8,e; in An (cf. I.A.6) which
we denote by the 'displacement vector' dP'. Since An may be covered by
one coordinate neighborhood Rn with local coordinates ul, ..., un,
we may look upon dP' as the vector whose components are the
differentials dul, ..., dun. Moreover, the ei are the natural basis vectors
alaut (i = 1, ..., n). Now, in affine space it is not necessary to introduce
the concept of covariant differential, and so the differential dei is a
vectorial 1-form for each i, and we may write

Differentiating the equations
ei . e, =
and applying (1.9.23) we obtain the first of equations (1.9.15) (cf. I.G).
The remaining formulae follow from those in 5 1.8 as well as (1.9.15).
We remark that the tensor RUk, = gimejk, satisfies the relations
(1 -9.19) - (1 -9.21).
The forms 8, and 841 are determined by the Riemannian metric
of the manifold. If we are given two such metrics dF2 and df2 in the local