3 Riemannian Geometry

Let us now consider a small region *M* of space on which it is possible to
put coordinates, say
(*x*_{1}, *x*_{2}, *x*_{3}). Moreover, any other choice of
coordinates, say
(*y*_{1}, *y*_{2}, *y*_{3}) is differentiable (to any order) with
respect to the given choice, that is to say the *y*_{i} are
differentiable functions of the *x*_{i}, and vice versa. Then we have
the non-singularity of the *Jacobian matrix*

With respect to the given choice of coordinates
(*x*_{1}, *x*_{2}, *x*_{3}) a vector
can be represented by a column vector
**a** = (*a*_{1}, *a*_{2}, *a*_{3})^{t}. A basis for the tangent vector space at all
points is given by the standard basis
{*e*^{1}, *e*^{2}, *e*^{3}} of .
We are also interested in tensor fields, which are sums of terms of the
form
*f*_{i1, i2,..., ir}*e*^{i1} *e*^{i2} ^{ ... } *e*^{ir} where
*f*_{i1, i2,..., ir} is a differentiable function.
The number *r* is called the order of the corresponding tensor. A tensor
of order 0 is thus just a differentiable function and one of order 1 is
a (tangent) vector field.

In another system of coordinates
(*y*_{1}, *y*_{2}, *y*_{3}) the vector
is given by the column vector
*J*(*x*, *y*)^{ . }**a**. This can be
extended to tensors in an obvious way. With this understanding we now
fix a choice of coordinates and a corresponding representation of
tangent vectors as column vectors.

In order to measure the magnitude and angle of velocities we are
given a symmetric positive definite bilinear pairing < , > on the tangent
space at each point. With respect to a given representation of tangent
vectors as column vectors, this is given by a positive definite
symmetric matrix
(*g*_{ij})_{i, j = 1}^{i, j = 3} of functions on the space.
Moreover, we further assume that the matrix entries *g*_{ij} are
differentiable (to any order) functions of the coordinates.
Such a pairing is called a *Riemannian metric*. Having prescribed this we
have a (local) Riemannian manifold.

The pairing < , > also extends to pairing on the various types of
tensors by induction on the order. In fact we can pair a tensor of order
*r* with a tensor of order *s* to obtain a tensor of order | *r* - *s*|.

For any vector field *V* we have learned in vector calculus to
compute the gradient
(*V*) with respect to a vector
at a point *p*. We have also learned about the gradient of a
function *f*. These give an example of a derivation; in other words we
have the Liebnitz rule

(*f*^{ . }*V*) = (*f* )^{ . }*A*(*p*) + *f* (*p*)^{ . }(*A*)

Now we can clearly extend this to all tensors by applying the Liebnitz
rule again as follows,
(*A* *B*) = (*A*) *B*(*p*) + *A*(*p*) (*B*)

Finally, we also have the linearity relation
= *a* + *b*

Now we can also consider other derivations
Given a vector field *V* we can apply the existence theorem for ordinary
differential equations to construct a 1-parameter flow on our
space *M*. This satisfies
= `o` and
fixes all points. Finally, we have
*d*(*p*)/*dt* = *V*((*p*)); which
is the ordinary differential equation we have solved. For any tensor
field *A*, the following limit exists by differentiability of the
quantities involved

While the gradient of a function is independent of a choice of
coordinates because of the interpretation as Lie derivative, the
gradient of a tensor of order at least 1 is clearly dependent upon the
choice of coordinates, thus it is natural to ask for a derivation *D*
that is in some sense ``canonical''. This is provided by the condition
that the inner product form < , > has no derivative,

()() = ()()

There is then a unique such derivation and thus it is independent of any
choice of coordinates.
In geometric terms, the usual gradient
(*A*) measures
the deviation of
*A*(*p* + *t*) from a copy of *A*(*p*) moved to the
point
*p* + *t*. However, this motion requires a notion of parallel
transport or rigid motion--which may be different for our geometry from
the Euclidean one provided by the coordinates; the Riemannian metric
provides the correct notion of angles and distances and hence of rigid
motion. Thus *D* provides the ``corrected'' gradient.

Now it is conceivable that there is a choice of coordinates in which the
()'s vanish or simplify. Thus we need to find some way of
checking this possibility. Riemann introduced the Riemannian curvature
as a measure of this. This was modified by Christoffel who defined the
curvature operator as follows. First of all for a vector field *V* and a
tensor field *A* let us define
*D*_{V}(*A*)(*p*) = *D*_{V(p)}(*A*). We then define
the operator *R* by

(*D*_{U}*R*)(*V*, *W*) + (*D*_{W}*R*)(*U*, *V*) + (*D*_{V}*R*)(*W*, *U*) = 0

We now consider the curvature as a function of a pair of tensors of
order 2
= ( - )

for skew-symmetric tensors of order 2. Then we have
< , > = ( < , > < , > - < , > < , > )

A comparison of the associated quadratic forms is then given by the
ratio
(,) = -

This is what is called the
An important result of Cartan and Hadamard (see [3]) states
that any Riemannian space *M* as above with the property that its
sectional curvature is constant can be identified with an open subset of
either the Euclidean space or Hyperbolic (Bolyai-Lobachevsky) space or
Projective space in an *isometric* way.

We will see in the next section how to apply this to the study of axiomatic geometries.