An open surface with
X-,
Y-, and
Z-contours shown.
In
mathematics, specifically in
topology, a
surface is a
two-dimensional topological manifold. The most familiar
examples are those that arise as the boundaries of solid objects in
ordinary three-dimensional
Euclidean
space R^{3} — for example, the surface
of a
ball. On the other hand, there are
surfaces which cannot be
embedded in
three-dimensional Euclidean space without introducing
singularities or intersecting itself —
these are the
unorientable
surfaces.
To say that a surface is "two-dimensional" means that, about each
point, there is a
coordinate patch on which a
two-dimensional
coordinate system
is defined.
For example, the surface of the Earth is (ideally) a two-dimensional sphere, and latitude and
longitude provide coordinates on it —
except at the International Date Line and the poles, where longitude is undefined.
This example illustrates that not all surfaces admits a single
coordinate patch. In general, multiple coordinate patches are
needed to cover a surface.
Surfaces find application in
physics,
engineering,
computer graphics, and many other
disciplines, primarily when they represent the surfaces of physical
objects. For example, in analyzing the
aerodynamic properties of an
airplane, the central consideration is the flow of
air along its surface.
Definitions and first examples
A
(topological) surface is a
Hausdorff topological space on which every point has
an open
neighbourhood
homeomorphic to some
open subset of the Euclidean plane
E^{2}. Such a neighborhood, together with
the corresponding homeomorphism, is known as a
(coordinate)
chart. It is through this chart that the neighborhood inherits
the standard coordinates on the Euclidean plane. This coordinates
are known as
local coordinates and these homeomorphisms
lead us to describe surfaces as being
locally
Euclidean.
More generally, a
(topological) surface with boundary is a
Hausdorff topological space in which every point has
an open
neighbourhood
homeomorphic to some
open subset of the
upper half-plane
H^{2}. These homeomorphisms are also known
as
(coordinate) charts. The boundary of the upper
half-plane is the
x-axis. A point on the surface mapped
via a chart to the
x-axis is termed a
boundary
point. The collection of such points is known as the
boundary of the surface which is necessarily a
one-manifold, that is, the union of closed curves. On the other
hand, a point mapped to above the
x-axis is an
interior point. The collection of interior points is the
interior of the surface which is always non-
empty. The closed
disk is a simple example of a surface
with boundary. The boundary of the disc is a circle.
The term
surface used without qualification refers to
surfaces without boundary. In particular, a surface with empty
boundary is a surface in the usual sense. A surface with empty
boundary and is compact is known as a 'closed' surface. The
two-dimensional sphere, the two-dimensional
torus, and the
real
projective plane are examples of closed surfaces.
The
Möbius strip is a surface with
only one "side". In general, a surface is said to be
orientable if it does not contain a homeomorphic copy of
the Möbius strip; intuitively, it has two distinct "sides". For
example, the sphere and torus are orientable, while the real
projective plane is not (because deleting a point or disk from the
real projective plane produces the Möbius strip).
In
differential and
algebraic geometry, extra structure is
added upon the topology of the surface. This added structures
detects
singularities, such
as self-intersections and cusps, that cannot be described solely in
terms of the underlying topology.
Extrinsically defined surfaces and embeddings
[[Image:Sphere_wireframe.svg|left|thumb|250px|A sphere can be
defined parametrically (by
x =
r sin
θ
cos
φ,
y =
r sin
θ sin
φ,
z =
r cos
θ) or implicitly
(by .)]]
Historically, surfaces were initially defined as subspaces of
Euclidean spaces. Often, these surfaces were the
locus of
zeros of certain functions, usually
polynomial functions. Such a definition considered the surface as
part of a larger (Euclidean) spaces, and as such was termed
extrinsic.
In the previous section, a surface is defined as a topological
space with certain property, namely Hausdorff and locally
Euclidean. This topological space is not considered as being a
subspace of another space. In this sense, the definition given
above, which is the definition that mathematicians use at present,
is
intrinsic.
A surface defined as intrinsic is not required to satisfy the added
constraint of being a subspace of Euclidean space. It seems
possible at first glance that there are surfaces defined
intrinsically that are not surfaces in the extrinsic sense.
However, the
Whitney embedding
theorem asserts that every surface can in fact be embedded
homeomorphically into Euclidean space, in fact into
E^{4}. Therefore the extrinsic and
intrinsic approaches turn out to be equivalent.
In fact, any compact surface that is either orientable or has a
boundary can be embedded in
E³; on the other hand,
the real projective plane, which is compact, non-orientable and
without boundary, cannot be embedded into
E³ (see
Gramain).
Steiner surfaces,
including
Boy's surface, the
Roman surface and the
cross-cap, are
immersions
of the real projective plane into
E³. These
surfaces are singular where the immersions intersect
themselves.
The
Alexander horned sphere
is a well-known
pathological embedding of the
two-sphere into the three-sphere.
The chosen embedding (if any) of a surface into another space is
regarded as extrinsic information; it is not essential to the
surface itself. For example, a torus can be embedded into
E³ in the "standard" manner (that looks like a
bagel) or in a
knotted manner (see figure). The two
embedded tori are homeomorphic but not
isotopic; they are topologically equivalent, but
their embeddings are not.
The
image of a continuous,
injective function from
R^{2} to higher-dimensional
R^{n} is said to be a
parametric surface. Such an image is
so-called because the
x- and
y- directions of the
domain
R^{2} are 2 variables that
parametrize the image. Be careful that a parametric surface need
not be a topological surface. A
surface of revolution can be viewed as
a special kind of parametric surface.
If
f is a smooth function from
R³ to
R whose
gradient is
nowhere zero, Then the
locus of
zeros of
f does define a
surface, known as an
implicit
surface. If the condition of non-vanishing gradient is
dropped then the zero locus may develop singularities.
Construction from polygons
Each closed surface can be constructed from an oriented polygon
with an even number of sides, called a
fundamental polygon of the surface, by
pairwise identification of its edges. For example, in each polygon
below, attaching the sides with matching labels (
A with
A,
B with
B), so that the arrows point
in the same direction, yields the indicated surface.
Image:SphereAsSquare.svg|
sphereImage:ProjectivePlaneAsSquare.svg|
real projective
planeImage:TorusAsSquare.svg|
torusImage:KleinBottleAsSquare.svg|
Klein bottle
Any fundamental polygon can be written symbolically as follows.
Begin at any vertex, and proceed around the perimeter of the
polygon in either direction until returning to the starting vertex.
During this traversal, record the label on each edge in order, with
an exponent of -1 if the edge points opposite to the direction of
traversal. The four models above, when traversed clockwise starting
at the upper left, yield
- sphere: A B B^{-1} A^{-1}
- real projective plane: A B A B
- torus: A B A^{-1} B^{-1}
- Klein bottle: A B A B^{-1}.
The expression thus derived from a fundamental polygon of a surface
turns out to be the sole relation in a
presentation of the
fundamental group of the surface with the
polygon edge labels as generators. This is a consequence of the
Seifert–van Kampen
theorem.
Gluing edges of polygons is a special kind of
quotient space process. The quotient concept
can be applied in greater generality to produce new or alternative
constructions of surfaces. For example, the real projective plane
can be obtained as the quotient of the sphere by identifying all
pairs of opposite points on the sphere. Another example of a
quotient is the connected sum.
Connected sums
The
connected sum of two surfaces
M and
N, denoted
M #
N, is
obtained by removing a disk from each of them and gluing them along
the boundary components that result. The boundary of a disk is a
circle, so these boundary components are circles. The
Euler characteristic \chi of
M
#
N is the sum of the Euler characteristics of the
summands, minus two:
- \chi(M \# N) = \chi(M) + \chi(N) - 2.\,
The sphere
S is an
identity element for the connected sum,
meaning that
S #
M =
M. This is
because deleting a disk from the sphere leaves a disk, which simply
replaces the disk deleted from
M upon gluing.
Connected summation with the torus
T is also
described as attaching a "handle" to the other summand
M.
If
M is orientable, then so is
T #
M. The connected sum is associative so the connected sum
of a finite number of surfaces is well-defined.
The connected sum of two real projective planes is the
Klein bottle. The connected sum of the real
projective plane and the Klein bottle is homeomorphic to the
connected sum of the real projective plane with the torus. Thus,
the connected sum of three real projective planes is homeomorphic
to the connected sum of the real projective plane with the torus.
Any connected sum involving a real projective plane is
nonorientable.
Classification of closed surfaces
The
classification theorem of closed surfaces states that
any closed surface is homeomorphic to some member of one of these
three families:
- the sphere;
- the connected sum of g tori, for g \geq 1;
- the connected sum of k real projective planes, for k
\geq 1.
The surfaces in the first two families are orientable. It is
convenient to combine the two families by regarding the sphere as
the connected sum of 0 tori. The number
g of tori involved
is called the
genus of the surface. Since the sphere and
the torus have Euler characteristics 2 and 0, respectively, it
follows that the Euler characteristic of the connected sum of
g tori is .
The surfaces in the third family are nonorientable. Since the Euler
characteristic of the real projective plane is 1, the Euler
characteristic of the connected sum of
k of them is
.
It follows that a closed surface is determined, up to
homeomorphism, by two pieces of information: its Euler
characteristic, and whether it is orientable or not. In other
words, Euler characteristic and orientability completely classify
closed surfaces up to homeomorphism.
Relating this classification to connected sums, the closed surfaces
up to homeomorphism form a
monoid with
respect to the connected sum. The identity is the sphere. The real
projective plane and the torus generate this monoid. In addition,
there is a relation
P #
P #
P =
P #
T – geometrically, connect sum with a torus
(# T) adds a handle with both ends attached to the same side of the
surface, while connect sum with a Klein bottle (# K = # P # P) adds
a handle with the two ends attached to opposite sides of the
surface; in the presence of a projective plane, the surface is not
orientable (there is no notion of side), so there is no difference
between attaching a torus and attaching a Klein bottle, which
explains the relation.
There are a number of proofs of this classification; most commonly,
it relies on the difficult result that every compact 2-manifold is
homeomorphic to a
simplicial
complex.
A closely related example to the classification of compact
2-manifolds is the classification of compact
Riemann surfaces, i.e., compact complex
1-manifolds. (Note that the 2-sphere, and the tori are all
complex manifolds, in fact
algebraic varieties.) Since every complex
manifold is orientable, the connected sums of projective planes do
not qualify. Thus compact Riemann surfaces are characterized
topologically simply by their genus. The genus counts the number of
holes in the manifold: the sphere has genus 0, the one-holed torus
genus 1, etc.
Surfaces in geometry
Polyhedra, such as the boundary of a
cube, are among the first surfaces encountered
in geometry. It is also possible to define
smooth
surfaces, in which each point has a neighborhood
diffeomorphic to some open set in
E². This elaboration allows
calculus to be applied to surfaces to prove many
results.
Two smooth surfaces are diffeomorphic if and only if they are
homeomorphic. (The analogous result does not hold for
higher-dimensional manifolds.) Thus
closed surfaces are classified up to
diffeomorphism by their Euler characteristic and
orientability.
Smooth surfaces equipped with
Riemannian metrics are of fundational
importance in
differential
geometry. A Riemannian metric endows a surface with notions of
geodesic,
distance,
angle, and area. It also gives rise to
Gaussian curvature, which
describes how curved or bent the surface is at each point.
Curvature is a rigid, geometric property, in that it is not
preserved by general diffeomorphisms of the surface. However, the
famous
Gauss-Bonnet theorem for
closed surfaces states that the integral of the Gaussian curvature
K over the entire surface
S is determined by the
Euler characteristic:
- \int_S K \; dA = 2 \pi \chi(S).
This result exemplifies the deep relationship between the geometry
and topology of surfaces (and, to a lesser extent,
higher-dimensional manifolds).
Another way in which surfaces arise in geometry is by passing into
the complex domain. A complex one-manifold is a smooth oriented
surface, also called a
Riemann
surface. Any complex nonsingular
algebraic curve viewed as a real manifold is
a Riemann surface.
Every closed orientable surface admits a complex structure. Complex
structures on a closed oriented surface correspond to
conformal equivalence classes of
Riemannian metrics on the surface. One version of the
uniformization theorem (due to
Poincaré) states that any
Riemannian metric on an oriented,
closed surface is conformally equivalent to an essentially unique
metric of
constant curvature.
This provides a starting point for one of the approaches to
Teichmüller theory, which
provides a finer classification of Riemann surfaces than the
topological one by Euler characteristic alone.
A
complex surface is a complex two-manifold and thus a
real four-manifold; it is not a surface in the sense of this
article. Neither are algebraic curves or surfaces defined over
field other than the complex
numbers.
See also
References
External links