differential geometry, branch of geometry geometry [Gr.,=earth measuring], branch of mathematics concerned with the properties of and relationships between points, lines, planes, and figures and with generalizations of these concepts.
..... Click the link for more information. in which the concepts of the calculus calculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit —the notion of tending toward, or approaching, an ultimate value.
..... Click the link for more information. are applied to curves, surfaces, and other geometric entities. The approach in classical differential geometry involves the use of coordinate geometry (see analytic geometry analytic geometry, branch of geometry in which points are represented with respect to a coordinate system, such as Cartesian coordinates , and in which the approach to geometric problems is primarily algebraic.
..... Click the link for more information. ; Cartesian coordinates Cartesian coordinates (kärtē`zhən)
..... Click the link for more information. ), although in the 20th cent. the methods of differential geometry have been applied in other areas of geometry, e.g., in projective geometry projective geometry, branch of geometry concerned with those properties of geometric figures that remain invariant under projection. The basic elements are points, lines, and planes, and the following statements are usually taken as assumptions: (1) two points lie in
..... Click the link for more information. .

The Analysis of Curves

If a point r moves along a curve at arc length s from some fixed point, then t = dr/ds is a unit tangent vector to the curve at r. The normal vector n is perpendicular to the curve at the point and indicates the direction of the rate of change of t, i.e., the tendency of r to bend in the plane containing both r and t, and the binormal vector b is perpendicular to both t and n and indicates the tendency of the curve to twist out of the plane of t and n.

These three vectors are related by the three formulas of the French mathematician Jean Frédéric Frenet, which are fundamental to the study of space curves: dt/ds = &kgr;n; dn/ds = −&kgr;t + τb; db/ds = −τn, where the constants &kgr; and τ are the curvature and the torsion of the curve, respectively. Of special interest are the curves called evolutes and involutes; the evolute of a curve is another curve whose tangents are the normals to the original curve, and an involute of a curve is a curve whose evolute is the given curve.

The Analysis of Surfaces

In the analysis of surfaces, points on a surface may be described not only with respect to the three-dimensional coordinates of the space in which the surface is considered but also with respect to an intrinsic coordinate system defined in terms of a system of curves on the surface itself. The curves on the surface that locally represent the shortest distances between points on the surface are called geodesics; geodesics on a plane are straight lines. Tangent and normal vectors are also defined for a surface, but the relationships between them are more complex than for a space curve (e.g., a surface has a whole circle of unit vectors tangent to it at a given point).

The results of the theory of surfaces are expressed most easily in the notation of tensors tensor, in mathematics, quantity that depends linearly on several vector variables and that varies covariantly with respect to some variables and contravariantly with respect to others when the coordinate axes are rotated (see Cartesian coordinates ).
..... Click the link for more information. . It is found that the total, or Gaussian, curvature of a surface is a bending invariant, i.e., an intrinsic property of the surface itself, independent of the space in which the surface may be considered. Of particular importance are surfaces of constant curvature; planes, cylinders, cones, and other so-called developable surfaces have zero curvature, while the elliptic and hyperbolic planes of non-Euclidean geometry non-Euclidean geometry, branch of geometry in which the fifth postulate of Euclidean geometry, which allows one and only one line parallel to a given line through a given external point, is replaced by one of two alternative postulates.
..... Click the link for more information. are surfaces of constant positive and negative curvature, respectively.

Development of Differential Geometry

Differential geometry was founded by Gaspard Monge and C. F. Gauss in the beginning of the 19th cent. Important contributions were made by many mathematicians during the 19th cent., including B. Riemann, E. B. Christoffel, and C. G. Ricci. This work was collected and systematized at the end of the century by J. G. Darboux and Luigi Bianchi. The importance of differential geometry may be seen from the fact that Einstein's general theory of relativity relativity, physical theory, introduced by Albert Einstein, that discards the concept of absolute motion and instead treats only relative motion between two systems or frames of reference.
..... Click the link for more information. is formulated entirely in terms of the differential geometry, in tensor notation, of a four-dimensional manifold combining space and time.

differential geometry

Field of mathematics in which methods of calculus are applied to the local geometry of curves and surfaces (i.e., to a small portion of a surface or curve around a point). A simple example is finding the tangent line on a two-dimensional curve at a given point. Similar operations may be extended to calculate the curvature and length of a curve and to analogous properties of surfaces in any number of dimensions.