Surface

One of the basic concepts in geometry. The definitions of a surface in various fields of geometry differ substantially.

In elementary geometry, one considers planes, multi-faced surfaces, as well as certain curved surfaces (for example, spheres). Each curved surface is defined in a special way, very often as a set of points or lines. The general concept of surface is only explained, not defined, in elementary geometry: One says that a surface is the boundary of a body, or the trace of a moving line, etc.

In analytic and algebraic geometry, a surface is considered as a set of points the coordinates of which satisfy equations of a particular form (see, for example, Surface of the second order; Algebraic surface).

In three-dimensional Euclidean space $ E ^ {3} $, a surface is defined by means of the concept of a surface patch — a homeomorphic image of a square in $ E ^ {3} $. A surface is understood to be a connected set which is the union of surface patches (for example, a sphere is the union of two hemispheres, which are surface patches).

Usually, a surface is specified in $ E ^ {3} $ by a vector function

$$ \mathbf r = \mathbf r ( x( u , v), y( u , v), z( u , v)), $$

where $ 0 \leq u , v \leq 1 $, while

$$ x = x( u, v),\ \ y = y( u, v),\ \ z = z( u, v) $$

are functions of parameters $ u $ and $ v $ that satisfy certain regularity conditions, for example, the condition

$$ \mathop{\rm rank} \left \| \begin{array}{lll} x _ {u} ^ \prime &y _ {u} ^ \prime &z _ {u} ^ \prime \\ x _ {v} ^ \prime &y _ {v} ^ \prime &z _ {v} ^ \prime \\ \end{array} \right \| = 2 $$

(see also Differential geometry; Theory of surfaces; Riemannian geometry).

From the point of view of topology, a surface is a two-dimensional manifold.

References

[a1]	J.J. Stoker, "Differential geometry" , Wiley (Interscience) (1969)
[a2]	J.A. Thorpe, "Elementary topics in differential geometry" , Springer (1979) MR0528129 Zbl 0404.53001