DIFFERENTIAL GEOMETRY

field of mathematics in which the calculus is applied to geometry. The type of this geometry called local deals mainly with properties in a limited domain around a point. It was thoroughly investigated in the 19th century after it was initiated in the 18th century by the European mathematicians Leonhard Euler and Gaspard Monge. With it are connected the names of Carl Friedrich Gauss, Pierre-Ossian Bonnet, Jean-Frdric Frenet, and Eugenio Beltrami. Considerations of finite parts of surfaces, or surfaces (and curves) as a whole, are the concern of integral geometry, or global geometry, which has been further developed in the 20th century by Wilhelm Blaschke and others. This specialty includes, for example, theorems on closed geodesic lines on surfaces, especially ovaloids (egglike surfaces). A simple example of the techniques of differential geometry is the determination of the tangent to a plane curve at some chosen point on the curve. The procedure is equivalent to selectingfrom all the lines passing through the chosen pointthat line which has the same slope as the curve at that point. The methods of analytic geometry make it possible to write algebraic equations for the curve and for a line passing through the desired point and any nearby point on the curve; the methods of the calculus identify the line that has the required slope. Similar operations may be extended to the calculation of the curvature and length of arc of curves and of analogous properties of surfaces in spaces of any number of dimensions. field of mathematics in which the calculus is applied to geometry. Differential geometry has its origin in the discovery in the 17th century of the infinitesimal calculus, one part of mathematics that deals with limits. The concept of the derivative of a function is essentially identical with that of the tangent line or slope of a curve, and the integral of a function can be geometrically interpreted as the area under a curve. The geometry of curves and surfaces in space was studied as an application of the calculus, leading to various notions of curvature. Among the main contributors were Leonhard Euler and Gaspard Monge. In this article there follows a treatment of differential geometry topics in which, after introductory material, consideration is given to manifolds and tensor bundles, operations on tensor fields, connections, interplay between local and global properties, the Gauss-Bonnet formula, elliptic operators, and, finally, modern development in surface theory. The equations of a surface S in a Euclidean space with the coordinates x1, x2, x3 can be given in parametric form (see 268). At the point (u1, u2) on the surface the components of a normal vector of unit length can also be given (see 269). The geometrical properties of S are then completely described by two quadratic differential forms (see 270), referred to respectively as the first and second fundamental forms of S. Gauss emphasized the importance of the properties of S that depend only on the first fundamental form, such as the length of a curve, the area of a domain, the geodesics (i.e., curves of shortest length), and the Gaussian curvature. Another German mathematician, Berhard Riemann, founded in 1854 what is now known as Riemannian geometry; that is, the geometry in a space of n dimensions with the coordinates ua, a = 1, . . . , n, based on a quadratic differential form (see 271). This geometry is of general form and includes as special cases the non-Euclidean geometries. It serves as a model of the physical universe in Einstein's general theory of relativity (see Riemannian geometry). Modern differential geometry stems from the basis that the objects of study are a class of spaces called manifolds equipped with additional structures. The main problems deal with the global properties of manifolds; i.e., properties that arise only when the manifolds are looked on as a whole. In global differential geometry topology is a major tool. Additional reading Treatments include Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, 2 vol. (196369); J.J. Stoker, Differential Geometry (1969, reissued 1989); Frank W. Warner, Foundations of Differentiable Manifolds and Lie Groups (1971, reissued 1983); Michael Spivak, A Comprehensive Introduction to Differential Geometry, 2nd ed., 5 vol. (1979); Marcel Berger and Bernard Gostiaux, Differential Geometry: Manifolds, Curves, and Surfaces (1988; originally published in French, 1972); Robert Wasserman, Tensors and Manifolds (1992); William M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed. (1986); Heinz Hopf, Differential Geometry in the Large, 2nd ed. (1989); and J.W. Bruce and P.J. Giblin, Curves and Singularities, 2nd ed. (1992). An updated version of a classic collection, of interest to theoretical physicists, is S.S. Chern (ed.), Global Differential Geometry (1989). Coverage of complex manifolds is found in S.S. Chern, Complex Manifolds Without Potential Theory, 2nd ed. (1979); Kunihiko Kodaira, Complex Manifolds and Deformation of Complex Structures, trans. from Japanese (1986); and F. Hirzebruch, Topological Methods in Algebraic Geometry, 3rd ed. (1966, reprinted with corrections 1978; originally published in German, 2nd ed., 1962). The K-theory and the index theorem are examined in M.F. Atiyah, K-theory (1967, reprinted 1989); Richard S. Palais (ed.), Seminar on the AtiyahSinger Index Theorem (1965); and Michael F. Atiyah and I.M. Singer, The Index of Elliptic Operators, IV, Annals of Mathematics, vol. 88 and 93 (1968, 1971). S.N.M. Ruijsenaars, Index Theorems and Anomalies: A Common Playground for Mathematicians and Physicists, CWI Quarterly, 3(1):319 (1990), introduces and connects the AtiyahSinger index theorem and anomaly theory. Papers on minimal varieties include Robert Osserman, A Proof of the Regularity Everywhere of the Classical Solution to Plateau's Problem, Annals of Mathematics, 91:550569 (1970); and E. Bombieri, E. de Giorgi, and E. Giusti, Minimal Cones and the Bernstein Problem, Inventiones Mathematicae, 7:243268 (1969). An additional collection of papers includes Themistocles M. Rassias (ed.), The Problem of Plateau (1992). S.S. Chern The Editors of the Encyclopdia Britannica