Page 69 - Curvature and Homology

EXERCISES 5 1

2. Let 0 be an arbitrary point of M and {y} the family of closed parametrized
curves on M with 0 as origin. The map

associates with each y E iY} an affine transformation T, of the tangent space at 0.
These transformations fbrm a group denoted by Ho-called the holonomy p p
at 0. The restricted holonomy group Hi consisting of the afKne linear maps Ti
is similarly defined. Show that the group Ho when considered as an abstract
group is independent of the choice of 0.
Hint: M is arcwise connected.
3. An affine connection is called a metrical connection if its restricted holonomy
group leaves invariant a positive definite quadratic form. Let M be an affinely
connected manifold with a metrical connection and assume that the scalar
product of two vectors is defined at some point 0 of M. Show that the scalar
product may be defined everywhere on M.
Hint: Let P be an arbitrary point of M, C a parametrized curve joining 0
and P and T& the affine linear map from To to I;D along C. Define the scalar
product at P by

and show that this definition is independent of the choice of C.
4. Show that the Levi Civita connection is a metrical connection.
5. Establish the equations (1.9.15).
One may proceed as follows: Develop the frames along C into affine space An.
Let X(to) and Y(to) be two vectors at C(to) and X'(to), Y1(t0) the corresponding
vectors at C'(to). Define a scalar product at C1(t0) by

By identifying An with one of its tangent spaces, a scalar product is defined in An.
From G.3, this scalar product is independent of the choice of to. In this way,
it follows that the orthonormal frames along C can be developed into An in
such a way that

where

The equations (1.9.15) follow by differentiating the last relation and
applying 1.F.