Manifold Atlas:Definition of “manifold”
1 Introduction
This page defines the term “manifold” as used in the Manifold Atlas. We assume that all manifolds are of a fixed dimension n.
Definition 1.1.
An n-dimensional manifold is a second countable, Hausdorff space for which every point
has a neighbourhood
homeomorphic to an open subset of
.
- The interior of
, denoted
, is the subset of points for which
is an open subset of
.
- The boundary of
, written
, is the complement of the interior of
.
-
is called closed if
is compact and
is empty.
A manifold as above is often called a topological manifold for emphasis or clarity.
Typically, but not necessarly, the word “manifold” will mean "topological manifold with extra structure", be it piecewise-linear, smooth, complex, symplectic, contact, Riemannian, etc. The extra structure will be emphasised or suppressed in notation and vocabulary as is appropriate. We briefly review some common categories of manifolds below.
2 Atlases of charts
We give a unified presentation of the definition of piecewise linear, smooth and complex manifolds. In the complex case, we assume that the dimension is even and that the dimension is even.
Recall that a chart on a topological manifold is a homeomporphism
from an open subset
of
to an open subset
of
. The transition function defined by two charts
and
is the homeomorphism
![\displaystyle \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).](/images/math/4/8/3/483e06ab62bb8088cf605417dac0a64c.png)
An atlas for is a collection of charts
such that the
cover
.
Let denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate
maps. That is we require every
to be either piecewise linear, smooth of class
or holomorphic. An atlas is a
Atlas if every transition function defined by the that atlas is a
function.
atlases are compatible if their union again forms a
atlas and by Zorn's Lemma each
atlas defines a unique maximal
atlas.
Definition 2.1. A -manifold
is a manifold
together with a maximal
atlas
.
![\Cat](/images/math/e/7/9/e7957168af2bb2dd58fbee67a6818dab.png)
![(M, A) \cong (N, B)](/images/math/2/c/e/2cebb92ef25cbd6106e623d3ddc231d8.png)
![f: M \cong N](/images/math/9/2/7/92713d356f50f96864fd7942e1c96517.png)
![\Cat](/images/math/e/7/9/e7957168af2bb2dd58fbee67a6818dab.png)
![A](/images/math/b/8/9/b8921ca1d75b852da96e95cda4aafeb8.png)
![B](/images/math/d/e/8/de80133e771f3ffb043b3ca894db2ccb.png)
3 Riemannian Manifolds
A Riemannian metric on a smooth manifold
is a smooth family of scalar products
![\displaystyle g_x : T_xM \times T_xM \longmapsto \Rr](/images/math/2/6/4/2647874adec97375b62c7e235c772c42.png)
defined on the tangent spaces for each
in
. This means that for each pair of smooth vector fields
and
on
the map
![\displaystyle M \to \Rr, ~~~ x \longmapsto g_x(v_1(x),v_2(x))](/images/math/6/6/a/66aa28e9d91def5edd6540acd367b0e0.png)
is smooth.
Definition 3.1.
A Riemannian manifold is a smooth manifold
together with a Riemannian metric
.
An isometry between Riemannian manifolds is a diffeomorphism whose differential preserves the metric .