2.7 KiB
Differential manifolds
In the following sections of differential geometry we make use of the Einstein summation convention introduced in vector analysis and \mathbb{K} = \mathbb{R}
or \mathbb{K} = \mathbb{C}.
Definition
Differential geometry is concerned with differential manifolds, smooth continua that are locally Euclidean.
Definition 1: let
n \in \mathbb{N}
, a $n$-dimensional differential manifold is a Hausdorff (T2) spaceM
furnished with a family of smooth diffeomorphisms\phi_\alpha: \mathscr{D}(\phi_\alpha) \to \mathscr{R}(\phi_\alpha)
with\mathscr{D}(\phi_\alpha) \subset\mathrm{M}
and\mathscr{R}(\phi_\alpha) \subset E
, with the following axioms
\mathscr{D}(\phi_\alpha)
is open and\bigcup_{\alpha \in \mathbb{N}} \mathscr{D}(\phi_\alpha) =\mathrm{M}
,- if
\Omega = \mathscr{D}(\phi_\alpha) \cap \mathscr{D}(\phi_\beta) \neq \empty
then\phi_\alpha(\Omega), \phi_\beta(\Omega) \subset E
are open sets and\phi_\alpha \circ \phi_\beta^{-1}, \phi_\beta \circ \phi_\alpha
are diffeomorphisms,- the atlas
\mathscr{A} = \{(\mathscr{D}(\phi_\alpha), \phi_\alpha)\}
is maximal.with
E
a $n$-dimensional Euclidean space.
The last axiom ensures that any chart is tacitly assumed to be already contained in the atlas.
Coordinate transformations
Definition 2: let
p,q \in \mathrm{M}
be points on the differential manifold and let\psi: \mathscr{D}(\psi) \to\mathrm{M}: p \mapsto \psi(p) \overset{\text{def}}{=} q
be a transformation fromp
toq
on the manifold, we define two diffeomorphisms
\phi_\alpha: \mathscr{D}(\phi_\alpha) \to \mathscr{R}(\phi_\alpha): p \mapsto \phi_\alpha(p) \overset{\text{def}}{=} x,
\phi_\beta: \mathscr{D}(\phi_\beta) \to \mathscr{R}(\phi_\beta): q \mapsto \phi_\beta(q) \overset{\text{def}}{=} y,
with
\mathscr{D}(\phi_{\alpha,\beta}) \subset\mathrm{M}
and\mathscr{R}(\phi_{\alpha,\beta}) \subset E
. Then we have a coordinate transformation given by
\phi_{\alpha \beta}^\psi = \phi_\beta \circ \psi \circ \phi_\alpha^{-1}: x \mapsto y,
then
\phi_{\alpha \beta}^\psi
is an active transformation ifp \neq q
and\phi_{\alpha \beta}^\psi
is a passive transformation ifp = q
.
To clarify the definitions, a passive transformation corresponds only to a descriptive transformation. Whereas an active transformation corresponds to a transformation on the manifold M
.
A passive transformation may also be given directly by \phi_\beta \circ \phi_\alpha: x \mapsto y
since \psi = \mathrm{id}
in this case. Note that the definitions could also have been given by the inverse as the transformations are all diffeomorphisms.