Page 43 - Curvature and Homology

1.7. AFFINE CONNECTIONS 25
The covariant derivative of a tensor field being itself a tensor field, we
may speak of second covariant derivatives, etc., the result again being
a tensor field.
Since Euclidean space En, considered as a differentiable manifold,
is covered by one coordinate neighborhood, it is not essential from
our point of view to introduce the concept of covariant derivative.
In fact, the affine connection is defined by setting the rJ; equal to
zero. The underlying affine space An is the ordinary n-dimensional
vector space-the tangent space at each point P of En coinciding '
with An. Indeed, the linear map sending the tangent vector a/aud to,
the vector (0, ..., 0, 1,0, ..., 0) (1 in the zTh place) identifies the tangent
space Tp with An itself. Let P and Q be two points of An. A tangent
vector at P and one at Q are said to be parallel if they may be identified
with the same vector of An. Clearly, the concept of parallelism (of
tangent vectors) in An is independent of the curve joining them.
However, in general, this is not the case as one readily sees from the
differential geometry of surfaces in E3. We therefore make the following
definition:
Let C = C(t) be a piecewise differentiable curve in M. The tangent
vectors

are said to be parallel along C if the covariant derivative DX(t) of X(t)
vanishes in the direction of C, that is, if

A piecewise differentiable curve is called an auto-parallel curve, if its
tangent vectors are parallel along the curve itself.
The equations (1.7.14) are a system of n first order differential
equations, and so corresponding to the initial value X = X(to) at t = to
there is a unique solution. Geometrically, we say that the vector X(t,)
has been given a parallel displacement along C. Algebraically, the parallel
displacement along C is a linear isomorphism of the tangent spaces at
the points of C. By definition, the auto-parallel curves are the integral
curves of the system

Hence, corresponding to given initial values, there is a unique auto-