Page 50 - Curvature and Homology

32 I. RIEMANNIAN MANIFOLDS

By permuting the indices i, j, and k, two further equations are obtained:

We define the contravariant tensor field gjk by means of the equations

Adding (1.9.8) to (1.9.9) and subtracting (1.9. lo), one obtains after
multiplying the result by gjm and contracting

where

and

(Although the torsion tensor vanishes, it will be convenient in § 5.3
to have the formula (1.9.12)). Hence, since the torsion tensor vanishes
(condition (a)), the connection qk is given explicitly in terms of the
metric by formula (1.9.13). That the b',) transform as they should is an
easy exercise. This is the connection of Lewi Civita. We remark that
condition (b) says that parallel displacement is an isometry. This follows
since parallel displacement is an isomorphic linear map between tangent
spaces.
A Riemannian metric gives rise to a submanifold f) of the bundle of
frames over M. This is the bundle of all orthonormal frames over M.
An orthonormal frame at a point P of M is a set of n mutually per-
pendicular unit vectors in the tangent space at P. In this case, the
structural group of the bundle is the orthogonal group. A connection
defined by a paralleliaation of f) gives a parallel displacement which is an
isometry-the Levi Civt'ta connection being the only one which is torsion free.
If we denote by Bi, Bij, e4$, Sf(kl the restrictions of P, B;, q, Silk, to
the orthonormal frames (cf. § 1.8), then by 'developing' the frames along
a parametrized curve C into affine space An (see the following paragraph),
it can be shown that