In classical differential geometry, development refers to the simple idea of rolling one smooth

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

over another in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

. For example, the

tangent plane In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...

to a surface (such as the

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...

or the

cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an infin ...

) at a point can be rolled around the surface to obtain the tangent plane at other points.

Properties

The tangential contact between the surfaces being rolled over one another provides a relation between points on the two surfaces. If this relation is (perhaps only in a

local Local may refer to: Geography and transportation * Local (train), a train serving local traffic demand * Local, Missouri, a community in the United States * Local government, a form of public administration, usually the lowest tier of administrat ...

sense) a bijection between the surfaces, then the two surfaces are said to be developable on each other or ''developments'' of each other. Differently put, the correspondence provides an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...

, locally, between the two surfaces. In particular, if one of the surfaces is a plane, then the other is called a

developable surface In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature. That is, it is a surface that can be flattened onto a plane without distortion (i.e. it can be bent without stretching or compression). ...

: thus a developable surface is one which is locally isometric to a plane. The cylinder is developable, but the sphere is not.

Flat connections

Development can be generalized further using flat connections. From this point of view, rolling the tangent plane over a surface defines an

affine connection In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...

on the surface (it provides an example of

parallel transport In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...

along a

curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...

), and a developable surface is one for which this connection is flat. More generally any flat

Cartan connection In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the ...

on a manifold defines a development of that manifold onto the model space. Perhaps the most famous example is the development of

conformally flat A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. In practice, the metric g of the manifold M has to be conformal to the flat metric \eta, i.e., the ...

''n''-manifolds, in which the model-space is the ''n''-sphere. The development of a conformally flat manifold is a conformal

local diffeomorphism In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between Smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below. Form ...

from the

universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete spa ...

of the manifold to the ''n''-sphere.

Undevelopable surfaces

The class of double-curved surfaces (undevelopable surfaces) contains objects that cannot be simply unfolded (developed). Such surfaces can be developed only approximately with some distortions of linear surface elements (see the Stretched grid method)

References

*{{cite book , first = R.W. , last = Sharpe , title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program , publisher = Springer-Verlag, New York , year = 1997 , isbn = 0-387-94732-9 Differential geometry Connection (mathematics)

Properties

Flat connections

Undevelopable surfaces

See also

References