Development (differential geometry)
Encyclopedia
In classical differential geometry, development refers to the simple idea of rolling one smooth surface
over another in Euclidean space
. For example, the tangent plane to a surface (such as the sphere
or the cylinder
) at a point
can be rolled around the surface to obtain the tangent-plane at other points.
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
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
, locally, between the two surfaces.
In particular, if one of the surfaces is a plane, then the other is called a developable surface
: thus a developable surface is one which is locally isometric to a plane. The cylinder is developable, but the sphere is not.
Development can be generalized further using flat connections. From this point of view, rolling the tangent plane over a surface defines an affine connection
on the surface (it provides an example of parallel transport
along a curve
), and a developable surface is one for which this connection is flat.
More generally any flat Cartan connection
on a manifold
defines a development of that manifold onto the model space
. Perhaps the most famous example is the development of conformally flat
n-manifolds, in which the model-space is the n-sphere. The development of a conformally flat manifold is a conformal local diffeomorphism
from the universal cover of the manifold to the n-sphere.
Surface
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 R3 — for example, the surface of a ball...
over another in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. For example, the tangent plane to a surface (such as the sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
or the cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
) at a point
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
can be rolled around the surface to obtain the tangent-plane at other points.
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 property
In mathematics, a phenomenon is sometimes said to occur locally if, roughly speaking, it occurs on sufficiently small or arbitrarily small neighborhoods of points.-Properties of a single space:...
sense) a bijection
Bijection
A bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
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
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
, locally, between the two surfaces.
In particular, if one of the surfaces is a plane, then the other is called a developable surface
Developable surface
In mathematics, a developable surface is a surface with zero Gaussian curvature. That is, it is a "surface" that can be flattened onto a plane without distortion . Conversely, it is a surface which can be made by transforming a plane...
: thus a developable surface is one which is locally isometric to a plane. The cylinder is developable, but the sphere is not.
Development can be generalized further using flat connections. From this point of view, rolling the tangent plane over a surface defines an affine connection
Affine connection
In the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
on the surface (it provides an example of parallel transport
Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
along a curve
Curve
In mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
), and a developable surface is one for which this connection is flat.
More generally any flat Cartan connection
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 geometry of the principal bundle is tied to the...
on a manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
defines a development of that manifold onto the model space
Klein geometry
In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on X by a Lie group G, which acts as the symmetry group of the geometry.For background and motivation...
. Perhaps the most famous example is the development of conformally flat
Conformal geometry
In mathematics, conformal geometry is the study of the set of angle-preserving transformations on a space. In two real dimensions, conformal geometry is precisely the geometry of Riemann surfaces...
n-manifolds, in which the model-space is the n-sphere. The development of a conformally flat manifold is a conformal local diffeomorphism
Local diffeomorphism
In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a function between smooth manifolds that preserves the local differentiable structure. The formal definition of a local diffeomorphism is given below....
from the universal cover of the manifold to the n-sphere.