Differential geometry

(mathematics) The study of curves and surfaces using the methods of differential calculus.

A branch of mathematics that deals with intrinsic properties of curves and surfaces in three-dimensional euclidean space. The intrinsic properties are those which are independent of the geometrical objects orientation or location in space. The subject is also concerned with nets of curves and families of surfaces, these having wide application in the arts.

The space is referred to a rectangular cartesian coordinate system (x,y,z). A space curve may be defined by a pair of independent equations, f(x,y,z) = 0 and g(x,y,z) = 0, or, more meaningfully, by parametric equations, x = x(t), y = y(t), z = z(t). In this case the arc length between two points t₀ and t is given by Eq. (1).
1.

Obviously, if s is chosen as parameter, Eq. (2) holds. In this
2.
case dx/ds, dy/ds, dz/ds at a point P are the direction cosines of the tangent to the curve at P.

Consider three nearby points P, P₁, P₂. Through them, in general, one plane may be constructed. The limiting position of this plane as P₁ and P₂ approach P is the osculating plane of the curve at P. The limiting position of the circle through P, P₁, P₂ as P₁ and P₂ approach P is the osculating circle. Its center is the center of curvature and its radius is the radius of curvature p. The reciprocal of the radius of curvature κ is the curvature; its value is shown in expression (3). The perpendicular to the tangent
3.
at P in the osculating plane is the principal normal and the perpendicular at P to the osculating plane is the binormal.

A surface in three-dimensional euclidean space may be given by f(x,y,z) = 0 or, more conveniently, by Eqs. (4). For a fixed
4.
value of v, Eqs. (4) describe a curve in the surface, a u line, and similarly for a fixed value of u. Thus (u,v) are curvilinear coordinates of a point in the surface. The most important quantity in the study of surfaces is the arc length of a curve. This is given by Eq. (5), where E, F, G are functions of partial derivatives of (x,y,z).
5.
In the real domain E, G, and EG − F² are nonnegative and may be zero only at singular points of the surface, or at points where the matrix of the partial derivatives of (x,y,z) is of rank less than two. The right-hand side of Eq. (5) is called the first fundamental form of the surface, where E, F, G are the first fundamental quantities of the surface. All applicable surfaces have the same first fundamental form. Thus any cylinder or cone which is developable (applicable to a plane) has the fundamental form du² + dv², where (u,v) are rectangular cartesian coordinates. Under an arbitrary transformation of the surface coordinates (u,v) the fundamental quantities E, F, G transform linearly so that many problems in the study of surfaces reduce to the question whether a coordinate system exists on the surface in which E, F, G satisfy desired conditions. See also Coordinate systems; Riemannian geometry.