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An Introduction to Differentiable Manifolds and Riemannian Geometry
William M. Boothby
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eBook - PDF
An Introduction to Differentiable Manifolds and Riemannian Geometry
William M. Boothby
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An Introduction to Differentiable Manifolds and Riemannian Geometry
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1
INTRODUCTION
TO
MANIFOLDS
In
this
chapter,
we
establish
some
preliminary
notations
and
give
an
intuitive,
geometric
discussion
of
a
number
of
examples
of
manifolds-the
primary
objects
of
study
throughout
the
book.
Most
of
these
examples
are
surfaces
in
Euclidean
space;
for
these-together
with
curves
on
the
plane
and
in
space-were
the
original
objects
of
study
in
classical
differential
geometry
and
are
the
source
of
much
of
the
current
theory.
The
first
two
sections
deal
primarily
with
notational
matters
and
the
relation
between
Euclidean
space,
its
model
R",
and
real
vector
spaces.
In
Section
3
a
precise
definition
of
topological
manifolds
is
given,
and
in
the
remaining
sections
this
concept
is
illustrated.
1
Preliminary
Comments
on
R"
Let
R
denote
the
real
numbers
and
R"
their
n-fold
Cartesian
product
R-,
the
set
of
all
ordered
n-tuples
(x',
...
,
x")
of
real
numbers.
Individual
n-
tuples
may
be
denoted
at
times
by
a
single
letter.
Thus
x
=
(x',
...,
x"),
a
=
(a',
...,
a"),
and
so
on.
We
agree
once
and
for
all
to
use
on
R"
its
topology
as
a
metric
space
with
the
metric
defined
by
1