CHAPTER 2
Preliminaries Concerning Manifolds
This section is written to provide notation and to serve as brief introduction to
the
,j£fp-theory
of differential forms. The general references here are [8], [43], [51],
[27] and [35], [55].
2.1. Manifolds
While many geometric constructions in
Rn
can be transferred to the Riemann-
ian manifolds, the sometimes cumbersome technical details are often new and de-
sired. Many unfamiliar differences will be explicitly emphasized here. Those dif-
ferences sometimes only technical, sometimes delicate and important, are scattered
throughout the research journals. Although, most of these facts will be left un-
proven in this text, we state them clearly so that they are available for a routine
verification.
Our ambient space, subject to weakly differentiable deformations, will be an
oriented compact (without boundary) smooth Riemannian manifold X of dimension
n ^ 2.
2.1.1. Legitimate balls. Making precise estimates demands that we must
work with one atlas A consisting of a finite number of coordinate charts (Q, K) G *4,
where K : SI
Rn
is a ^°°-diffeomorphism of an open region ft C X onto
W1.
Let us choose and fix such an atlas A and call it the reference atlas. We then
introduce the so-called reliable radius of the manifold X. This is a positive number,
denoted by R = Rx, such that for 0 r Rx every pair of concentric geodesic
balls B(x, r) C B(x, Ar) C X fits in one coordinate region ft of the atlas A. We refer
to such B(x, r) as legitimate ball in X. The point to introducing this concept is that
estimates on a legitimate ball can be reduced equivalently to analogous estimates
in the Euclidean space. We mention now that the legitimate balls B = B(x,r) C X
share basic properties of the Euclidean balls. In particular,
(2.1) |B| ^ (diamB)n ^ |B|
Here the implied constant depends only on the manifold X.
2.1.2. Whitney covering. The familiar decomposition of an open set H c R n
into Whitney cubes can be adapted to manifolds. While cubes are perfect regions for
constructing various partitions of
Mn,
there are serious geometric and combinatorial
obstacles to do the same on manifolds. We shall work with the legitimate balls
instead of cubes. Since it is impossible to partition a manifold into mutually disjoint
balls, we will work with a finite covering in which the number of overlapping balls
depends only on the manifold X.
PROPOSITION
2.1. Given a non-empty open set Q £ X and its complement
¥ = X\Q. There exists a collection T {Bi,B2,...} of legitimate balls B^ C X such
that
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