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 [24].

Riemannian geometry is a non-Euclidean geometry that studies local properties of a smooth manifold.