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).
Copyright (c) 1995 Encyclopaedia Britannica, Inc. All Rights Reserved
Related Propaedia Topics:
Non-Euclidean geometry
Non-Euclidean geometry