Encyclopedia
In
mathematics, specifically in
topology, a
surface is a two-dimensional
manifold. The most familiar examples are those that arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space,
E3. On the other hand, there are also more exotic surfaces, that are so "contorted" that they cannot be embedded in three-dimensional space at all.
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 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 in general it is not possible to extend any one coordinate patch to the entire surface; surfaces, like manifolds of all dimensions, are usually constructed by patching together multiple coordinate systems.
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
surface with boundary is a
Hausdorff topological space in which every point has an open neighbourhood homeomorphic to some
open subset of the closed half space of
E2 . The neighborhood, along with the homeomorphism to Euclidean space, is called a
chart.
The set of points that have an open neighbourhood homeomorphic to
E2 is called the
interior of the surface; it is always non-empty. The complement of the interior is called the
boundary; it is a one-manifold, or union of closed curves. The simplest example of a surface with boundary is the closed disk in
E2; its boundary is a circle.
A surface with an empty boundary is called
boundaryless. A
closed surface is one that is boundaryless and compact. 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 .
More generally, it is common in differential and algebraic geometry to study surfaces with singularities, such as self-intersections, cusps, etc.
Extrinsically defined surfaces and embeddings
Historically, surfaces were originally defined and constructed not using the abstract,
intrinsic definition given above, but
extrinsically, as subsets of Euclidean spaces such as
E3.
Let
f be a continuous, injective function from
R2 to
R3. Then the image of
f is said to be a
parametric surface. A
surface of revolution can be viewed as a special kind of parametric surface.
On the other hand, suppose that
f is a smooth function from
R3 to
R whose
gradient is nowhere zero. Then the locus of
zeros of
f is said to be an
implicit surface. If the condition of non-vanishing gradient is dropped then the zero locus may develop singularities.
One can also define parametric and implicit surfaces in higher-dimensional Euclidean spaces
En. It is natural to ask whether all surfaces arise as subsets of some
En. The answer is yes; the Whitney embedding theorem, in the case of surfaces, states that any surface can be embedded homeomorphically into
E4. 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 and boundaryless, cannot be embedded into
E3. 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.
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 , so that the arrows point in the same direction, yields the indicated surface.
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:
- real projective plane:
- torus:
- Klein bottle: .
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.
Quotients and connected summation
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.
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
Euler characteristic of
M #
N is the sum of the Euler characteristics of the summands, minus two:
.
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 has the effect of attaching a "handle" to the other summand
M. If
M is orientable, then so is
T #
M. The connected sum can be iterated to attach any number
g of handles to
M.
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. Any connected sum involving a real projective plane is nonorientable.
Classification of closed surfaces
The
classification 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 ;
- the connected sum of k real projective planes, for .
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 2 - 2
g.
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 2 -
k.
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.
Surfaces in differential 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. Smooth surfaces equipped with Riemannian metrics are of foundational importance in differential geometry.
Minimal surfaces are surfaces that minimize the surface area for given
boundary conditions. Examples include soap films stretched across a wire frame,
catenoids and
helicoids.
Developable surfaces are surfaces that can be flattened to a plane without stretching; examples include the cylinder, the cone, and the torus.
Ruled surfaces are surfaces that have at least one straight line running through every point. Examples include the cylinder and the
hyperboloid of one sheet.
Surfaces in complex and algebraic geometry
Any
n-dimensional complex manifold is, at the same time, a real -dimensional real manifold. Thus any complex one-manifold is a topological surface. Any complex algebraic curve or real algebraic surface is also a topological surface, possibly with singularities.
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 fields other than the complex numbers.
See also
- Volume form, for volumes of surfaces in En
- Poincaré metric, for metric properties of Riemann surfaces
References
External links
