A locally Euclidean space with a differentiable structure. Let
be a topological Hausdorff space.
is known as a
locally Euclidean space
or as a
topological manifold
of dimension
if for each point
a neighbourhood
of
can be found that is homeomorphic to an open set of
.
The pair
,
where
is this homeomorphism, is known as a
local chart
of
at
.
Thus, to each point corresponds a selection of
real numbers
,
known as the
coordinates
of
in the chart
.
A family of charts
,
,
is known as an
-dimensional
-atlas
of
if a) the totality of all
covers
,
;
and b) for any
such that
,
the mapping
belongs to the
class of differentiability
;
is a differentiable mapping with non-vanishing Jacobian and
is known as a transformation of coordinates
from the chart
into the chart
.
Two
-atlases
are said to be
equivalent
if their union is again a
-atlas.
The set of
-atlases
is thus subdivided into equivalence classes, known as
-structures;
if
,
they are known as
differentiable
(or
smooth)
structures,
while if
they are known as
analytic structures.
The topological manifold
with a
-structure
is known as a
-manifold,
or as a
differentiable manifold of class
.
The concept of a differentiable structure may be introduced for an arbitrary set
by replacing the homeomorphisms
by bijective mappings on open sets of
;
here, the topology of the
-manifold
is described as the topology of the union,
constructed from an arbitrary atlas of the corresponding structure. In such a case
-dimensional
manifolds clearly have an
-dimensional
-structure.
Problems of analytical and algebraic geometry make it necessary to
consider in the definition of a differentiable structure not only the space
,
but also more general spaces, such as
or even
where
is a complete non-discretely normed field. Thus, if
,
the corresponding
-structure,
,
invariably proves to be a
-structure
and is called
complex-analytic,
or simply
complex,
while the corresponding differentiable manifold is known as a
complex manifold.
Such a manifold also carries a natural real
-structure.
Any
-manifold
contains a
-structure,
and there is a
-structure
on a
-manifold,
,
if
.
Conversely, any paracompact
-manifold,
,
may be provided with a
-structure
compatible with the given one, and this structure is unique,
up to an isomorphism (see below). It may happen, however, that a
-manifold
cannot be provided with a
-structure
(i.e. there exist non-smoothable manifolds, cf.
Non-smoothable manifold),
and even if it can be provided with such a structure,
the structure need not be unique. For example, the number
of
non-isomorphic
structures on the
-dimensional
sphere is:
 |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
8 |
9 |
10 |
11 |
12 |
 |
1 |
1 |
1 |
? |
1 |
1 |
28 |
2 |
8 |
6 |
992 |
1 |
|
Let
be a continuous mapping of
-manifolds
;
it is known as a
-morphism
(or as a
-mapping,
,
or as a
mapping of class
)
of differentiable manifolds if for any pair of charts
on
and
on
such that
,
the mapping
belongs to the class
.
A bijective mapping
such that it and
are
-mappings
is called a
-isomorphism
(or a
diffeomorphism of class
).
In such a case
and
and their determining
-structures
are said to be
-diffeomorphic.
A subspace
of an
-dimensional
-manifold
is called a
-submanifold
of dimension
in
if for any point
there exists a neighbourhood
of it and a chart
of the
-structure
such that
and
induces a homeomorphism of
onto the intersection of
with the closed subspace
;
in other words, there exists a chart with coordinates
such that
is defined by the relations
.
A mapping
is said to be a
-imbedding
if
is a
-submanifold
in
and if
is a
-diffeomorphism.
Any
-dimensional
-manifold
permits an imbedding in
and even in
.
Moreover, the set of such imbeddings is everywhere dense in the space of mappings
with respect to the compact-open topology. Thus, regarding a differentiable manifold as
a submanifold of a Euclidean space is one of the ways
of interpreting the theory of differentiable manifolds;
for example, the above theorems on
-structures
can be proved in this manner.
There are two fundamental problems in the topology of
differentiable manifolds (which is also referred to as
differential topology).
The first problem is the classification of differentiable
manifolds. There exist three main classes of differentiable
manifolds — closed (or compact) manifolds, compact manifolds with
boundary and open manifolds. Important invariants by
which differentiable manifolds are distinguished are the
homotopy type
and the
tangent bundle,
in particular the characteristic classes (cf.
Characteristic class).
Using these a classification of smooth structures
for simply-connected manifolds of given homotopy type
has been given. Another invariant — the
bordism
class of a differentiable manifold — was used in solving the generalized
Poincaré conjecture,
in the study of fixed points under the action of a group on
a manifold, etc. This involved the introduction of differentiable structures on
manifolds with boundary and of a smoothing apparatus. Finally, methods of
algebraic topology also proved useful in this context, since,
for example, they permitted to establish that any
-manifold
can be triangulated.
The second problem is the
classification of mappings
of differentiable manifolds. The first class to be considered
are immersions, which are a generalization of imbeddings; their classification
is reduced to a homotopy problem, as distinct from
imbeddings, which have not yet
(1987)
been completely classified (cf.
Topology of imbeddings),
and submersions, or fibrations, of one differentiable
manifold into another. In particular, the concept of
a transversal mapping along a submanifold plays an important role in problems of
stability
and in the study of typical singularities of mappings. The existence of
transversal mappings is ensured by theorems such as Sard's theorem (cf.
Sard theorem).
All this, and problems in differential dynamics, dealing with
the structure of various groups of diffeomorphisms (cf.
Diffeomorphism),
in particular of integral trajectories and singular points of vector
fields on differentiable manifolds (dynamical systems), as well as
the various equivalence relationships — isotopy, topological and
-conjugacy,
etc. — makes it necessary to study finite-dimensional spaces
together with arbitrary Banach (or Hilbert) spaces and to
determine corresponding differentiable structures. This implies finding additional conditions
that are reasonable from the point of view of applications, e.g.,
a differentiable manifold is separable if and only if the
coordinate transformations have a closed graph. In general,
infinite-dimensional manifolds provided with such a structure — known as
Banach
or
Hilbert manifolds,
respectively, manifolds of mappings of finite-dimensional manifolds being their typical
example — are a useful outcome of studies and geometrical
interpretation of problems of approximation of mappings (as in the
imbedding theorem above), in the analysis of loop spaces (a
suitable domain for the construction of Morse theory, cf.
Loop space),
etc.
Differentiable manifolds form a natural base for
developing differential geometry. Supplementary infinitesimal structures — orientation,
metric, connections, etc. — are introduced on differentiable manifolds,
after which a study is made of the objects which
are invariant with respect to the group of diffeomorphisms
which preserve the supplementary structure. Conversely, the use of a specific
structure permits one to study the structure of the differential
manifold itself. The simplest example is the expression of the
characteristic classes in terms of the curvature of
a differentiable manifold with a linear connection.