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
.
-
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 of
is even and that the boundary of
is empty.
Recall that a chart on a topological manifold is a homeomorphism
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 \circ \phi_\beta^{-1}|_{\phi_{\beta}(U_\alpha \cap U_\beta)} : \phi_\beta(U_\alpha \cap U_\beta) \longrightarrow \phi_\alpha(U_\alpha \cap U_\beta).](/images/math/0/f/e/0fe6c284d883a43d6e7bb31efa09b50c.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. An atlas is a
Atlas if every transition function defined by the that atlas is a
function: that is, we require every
to be either piecewise linear, smooth of class
or holomorphic. Two
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 .