The invariance of the [ds.sup.2] is a very important argument on behalf of Riemannian geometry
as the mathematical basis of General Relativity.
Yano, Integral Formulas in Riemannian Geometry
, Marcel Dekker, New York, 1970.
Various approaches to calculating the geometric analysis are then discussed, drawing on the fields of differential geometry, Riemannian geometry
, algebra, statistics, and computer science.
Chen: Riemannian geometry
of Lagrangian submanifolds,TaiwaneseJ.Math.,5(2001), No.
O' Neill, Semi Riemannian geometry
, Academic Press, New York, 1983.
Li, An Introduction to Riemannian Geometry
, Peking University Press, Beijing, China, 2002.
Many classical results from Riemannian geometry
have Lorentz counterparts.
BOOTHBY, An Introduction to Differentiable Manifolds and Riemannian Geometry
, Academic Press, New York, 1975.
While in the Riemannian geometry
, called elliptic geometry, the fifth Euclidean postulate is also invalidated as follows: there is no parallel to a given line passing through an exterior point.
(For applications of these methods in the context of Riemannian geometry
, see, e.g.
They cover the concentration of measure effects in quantum information, quantum error correction and fault-tolerant quantum computation, Riemannian geometry
of quantum computation, topological quantum information theory, quantum knots and mosaics, quantum knots and lattices as a blueprint for quantum systems that do rope tricks, and a Rosetta Stone for quantum mechanics with an introduction to quantum computation.
An important special case is when [F.sup.2] = [g.sub.ij] (x) [dx.sup.i][dx.sup.j] Historical developments have conferred the name Riemannian geometry
to this case while the general case, Riemannian geometry
without the quadratic restriction, has been known as Finsler geometry .
is a non-Euclidean geometry that studies local properties of a smooth manifold.