Last Updated：2022/12/24
A set S⊂ℝ³ is called a regular surface if each of its points has a neighborhood in S that is diffeomorphic to an open set in ℝ². That is, for every p∈S, there exists a neighborhood, V, of p in S, an open set U⊂ℝ², and a diffeomorphism 𝜎:U→V. Such a diffeomorphism 𝜎 is called a surface patch or a coordinate chart. A collection of surface patches that together cover all of the points of S is called an atlas for S.
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A
set
S⊂ℝ³
is
called
a
regular
surface
if
each
of
its
points
has
a
neighborhood
in
S
that
is
diffeomorphic
to
an
open
set
in
ℝ².
That
is,
for
every
p∈S,
there
exists
a
neighborhood,
V,
of
p
in
S,
an
open
set
U⊂ℝ²,
and
a
diffeomorphism
𝜎:U→V.
Such
a
diffeomorphism
𝜎
is
called
a
surface
patch
or
a
coordinate
chart.
A
collection
of
surface
patches
that
together
cover
all
of
the
points
of
S
is
called
an
atlas
for
S.
