RIEMANNIAN GEOMETRY

also called Elliptic Geometry, one of the non-Euclidean geometries that completely rejects the validity of Euclid's fifth postulate and modifies his second postulate. Simply stated, Euclid's fifth postulate is: through a point not on a given line there is only one line parallel to the given line. In Riemannian geometry, there are no lines parallel to the given line. Euclid's second postulate is: a straight line of finite length can be extended continuously without bounds. In Riemannian geometry, a straight line of finite length can be extended continuously without bounds, but all straight lines are of the same length. The tenets of Riemannian geometry, however, admit the other three Euclidean postulates (compare hyperbolic geometry). Although some of the theorems of Riemannian geometry are identical to those of Euclidean, most differ. In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant. In elliptic geometry, parallel lines do not exist. In Euclidean, the sum of the angles in a triangle is two right angles; in elliptic, the sum is greater than two right angles. In Euclidean, polygons of differing areas can be similar; in elliptic, similar polygons of differing areas do not exist. The first published works on non-Euclidean geometries appeared around 1830. Such publications were unknown to the German mathematician Bernhard Riemann who, in 1866, extended the concepts from two to three or more dimensions. Another German mathematician, Felix Klein, later discriminated between elliptical space (polar) and double-elliptical space (antipodal). also called elliptic geometry one of the non-Euclidean geometries, which completely reject the validity of Euclid's fifth postulate and modify his second postulate. Simply stated, Euclid's fifth postulate is: through a point not on a given line there is only one line parallel to the given line. In Riemannian geometry, there are no lines parallel to the given line. Euclid's second postulate is: a straight line of finite length can be extended continuously without bounds. In Riemannian geometry, a straight line of finite length can be extended continuously without bounds, but all straight lines are of the same length. The tenets of Riemannian geometry, however, admit the other three Euclidean postulates (compare hyperbolic geometry). Although some of the theorems of Riemannian geometry are identical to those of Euclidean, most differ. In Euclidean geometry, for example, two parallel lines are taken to be everywhere equidistant. In elliptic geometry, parallel lines do not exist. In Euclidean, the sum of the angles in a triangle is two right angles; in elliptic, the sum is greater than two right angles. In Euclidean, polygons of differing areas can be similar; in elliptic, similar polygons of differing areas do not exist. In a lecture entitled ber die Hypothesen, welche der Geometrie zu Grunde liegen (1854; On the Hypotheses Which Form the Foundations of Geometry), Bernhard Riemann developed a comprehensive view of geometry. With a thorough understanding of the limitations of Euclidean geometry, he formulated what is now known as double elliptic geometry, a type of elementary non-Euclidean geometry. In so doing, Riemann apparently was unaware that Nikolay Lobachevsky and Jnos Bolyai had already shown the possibility of devising a consistent geometry without the postulate of parallels on which Euclidean geometry is based (see non-Euclidean geometry: History of hyperbolic geometry). In effect, Riemann's work constituted an alternative to Lobachevsky's and Bolyai's systems of non-Euclidean geometry. Riemann's geometry played a fundamental role in the mathematical formulation of relativity theory. Basically, Riemannian geometry is concerned with the properties of a coordinate space (x1, . . . , xn) in which there is a nondegenerate quadratic differential form called the element of arc (see 311). This geometry reduces to Euclidean geometry if the element of arc takes the special form ds2 = (dx1)2 + . . . + (dxn)2. The two-dimensional case had been considered before Riemann by C.F. Gauss as the intrinsic geometry on a surface in ordinary Euclidean space. In pure mathematics, the differential form ds2 is generally supposed to be positive definite, an assumption that is essential to many of the important consequences. In applications to general relativity, however, in which the Riemannian space is the physical universe, ds2 is supposed to be hyperbolic; i.e., reducible to a sum of squares minus the square of a linear differential form. Additional reading Studies of Riemannian geometry include Luther Pfahler Eisenhart, Riemannian Geometry (1926, reissued 1964); Tullio Levi-Civita, The Absolute Differential Calculus (1926, reprinted 1977; originally published in Italian, 1925); C.E. Weatherburn, An Introduction to Riemannian Geometry and the Tensor Calculus (1938, reissued 1966); James Simons, Minimal Varieties in Riemannian Manifolds, Annals of Mathematics, 88:62105 (1968); H.M. Farkas and I. Kra, Riemann Surfaces, 2nd ed. (1991); Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian Geometry, 2nd ed. (1990); Frank Morgan, Riemannian Geometry: A Beginner's Guide (1993); and Manfredo Perdigo do Carmo, Riemannian Geometry (1992; originally published in Spanish, 2nd ed., 1988). Wilhelm Klingenberg, Riemannian Geometry (1982), covers geodesics. Complementary to this is John Stillwell, Geometry of Surfaces (1992). Robert Coquereaux and Arkadiusz Jadczyk, Riemannian Geometry, Fiber Bundles, Kaluza-Klein Theories, and All That . . . (1988), focuses on connections with particle physics. The Editors of the Encyclopdia Britannica