CHAPTER
I
ELEMENTARY DIFFERENTIAL
GEOMETRY
§1-§3. When a Euclidean space is stripped of its vector space structure and
only its differentiable structure retained, there are many ways of piecing together
domains of it in a smooth manner, thereby obtaining a so-called differentiable
manifold. Local concepts like a differentiable function and a tangent vector can
still be given a meaning whereby the manifold can be viewed "tangentially," that
is, through its family of tangent spaces as a curve in the plane is, roughly
speaking, determined
by its family of tangents. This viewpoint leads to the
study of tensor fields, which are important
tools in local and global differential
geometry. They form an algebra
(M),
the
mixed tensor algebra over the
manifold
M.
The alternate covariant tensor fields (the differential forms) form
a submodule
9t(M)
of
(M)
which inherits
a 'multiplication from
(M),
the
exterior multiplication. The resulting algebra
is called the Grassmann algebra
of
M.
Through the work of E. Cartan the Grassmann algebra with the exterior
differentiation
d
has become an indispensable tool for dealing with submanifolds,
these being analytically described by the zeros of differential forms. Moreover,
the pair
((M),
d)
determines the cohomology of
Al
via de Rham's
theorem,
which however will not be dealt with here.
§4-§8. The concept of an affine connection
was first defined by Levi-Civita
for Riemannian manifolds, generalizing significantly the notion of parallelism for
Euclidean spaces. On a manifold with a countable basis an affine connection always
exists (see the exercises following this chapter).
Given an affine connection on
a manifold
M
there is to each curve y(t) in
M
associated an isomorphism between
any two tangent spaces M,(,,) and
My(t,).
Thus, an affine connection makes it
possible to relate tangent spaces at distant points of the manifold. If the tangent
vectors of the curve y(t) all correspond under these isomorphisms we have the
analog of a straight line, the so-called geodesic. The theory of affine connections
mainly amounts to a study of the mappings Exp,:
M, - M
under which straight
lines (or segments of them) through the origin in the tangent space
M,
correspond
to geodesics through p in
M.
Each mapping Exp, is a diffeomorphism of a neigh-
borhood of 0 in
M,
into
M,
giving the so-called normal coordinates at p.
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