Riemannian geometry

Differential geometry of n-dimensional non-Euclidean space was developed by Riemann and is best known in its four-dimensional form that provided the basis of the general theory of relativity. Elementary examples of Riemann spaces include Euclidean space, spherical surfaces and hyperbolic spaces. 

The measurable distance between points inhnitely close together in three-dimensional Euclidean space is specihed as 

This space-time continuum differs from Euclidean space in that ds may be either positive or negative to differentiate between time-like and space-like 

The effect of curvature is that the simple differential form can only be valid in the immediate neighbourhood of any point. For a finite region of a curved surface ds is not invariant. It depends on the position in the surface and takes the form 

Geodesics are the curves that trace the shortest distance between two points, sufficiently close to each other. An n-dimensionaJ domain has the volume given by the integral 

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Bo] Bootbby, W.M., An Introduction to Differentiable Manifolds and Riemannian Geometry, Second Edition, Academic Press, Inc., Orlando, 1986. 

In Kirkwood s original formulation of the Fokker-Planck theory, he took into account the possibility that various constraints might apply, e.g., constant bond length between adjacent beads. This led to the introduction of a chain space of lower dimensionality than the full 3A-dimen-sional configuration space of the entire chain and it led to a complicated machinery of Riemannian geometry, with covariant and contravariant tensors, etc. 

This book has two primary aims. The first is to provide an accurate but accessible introduction to the theory of chemical and phase thermodynamics as first enunciated by J. Willard Gibbs. The second is to exhibit the transcendent beauty of the Gibbsian theory as expressed in the mathematical framework of Euclidean and Riemannian geometry. 

The general line-element expression (9.28) allows one to envision possible geometries with fto/i-Euclidean metric [i.e., failing to satisfy one or more of the conditions (9.27a-c)] or with variable metric [i.e., with a matrix M that varies with position in the space, M = M( i )> a Riemannian geometry that is only locally Euclidean cf. Section 13.1]. However, for the present equilibrium thermodynamic purposes (Chapters 9-12) we may consider only the simplest version of (9.28), with metric elements (R R,-) satisfying the Euclidean requirements (9.27a-c). 

G. Ruppeiner. Riemannian geometry in thermodynamic fluctuation theory. Rev. Mod. Phys. 67, 605-59 (1995). 

We now wish to introduce a still deeper form of geometry as first suggested by Bernhard Riemann (Sidebar 13.2). Riemann s formalism makes possible a distinction between the space of vectors whose metrical relationships are specified by the metric M and an associated linear manifold by which the vectors and metric are parametrized. Let be an element of a linear manifold (in general, having no metric character) that can uniquely identify the state of a collection of metrical objects X). The Riemannian geometry permits the associated metric M to itself be a function of the state,... 

The key feature of Riemannian geometry is the concept of a line element ds whose length is given by (Riemann s only equation in his 1854 Habilitationsvortrag)... 

In principle, the integrand in (13.10) might be evaluated with Taylor series expansions such as (12.96), based on successively higher derivatives of the initial state. In practice, however, direct experimental evaluation of the functional dependence of each My on path variables would be needed to evaluate C along extended paths. Further discussion of global curvature or other descriptors of the Riemannian geometry of real substances therefore awaits acquisition of appropriate experimental data, well beyond that required to describe individual points on a reversible path. 

A Riemann surface is a 2-dimensional compact differentiable surface, together with an infinitesimal element of length (see textbooks on differential and Riemannian geometry, for example, [Nak90]). The curvature K(x) at a point x is the coefficient a in the expansion ... 

Now, a theorem in Riemannian geometry tells us that locally any metric (7) with the correct signature can be rewritten as (6) by an appropriate change of coordinates. At different points we use different transformations of coordinates, but always end up with the Lorentz metric in the new coordinates. So the equation (8), when written in terms of the coordinates for which the metric looks like (7), must describe the trajectory in the gravitational field. This is the geodesic equation (sum over / , 7)... 

Since there is no reflection symmetry in the quaternion formulation, the reflected quaternion q 1 must be distinguishable from. The conjugate differential fine element to ds is ds = q dxVL. The product of the quaternion and conjugate quaternion line elements is then the real-number-valued element that corresponds to the squared differential element of the Riemannian geometry ... 

Vol. 509 D. E Blair, Contact Manifolds in Riemannian Geometry. VI, 146 pages. 1976. 

Attempts to accommodate the electromagnetic field in addition to the gravitational lead to a further generalization of the Riemannian geometry, for instance to the five-dimensional Kaluza metric or projective relativity of Veblen. 

To specify the directions of two different vectors at nearby points it is necessary to define tangent vectors at these points. Stated in different terms, at each point of space-time, known as the contact point, there is an associated tangent Minkowski space. The theory of these spaces together with the underlying space becomes a Riemannian geometry if a Euclidean metric is introduced in each tangent space by means of a differential quadratic form. ... 

This cone is real in the case of relativity theory, while the quadratic form gij is indefinite. Prom the point of view stressed by E. Cartan (bibb 1928,1) the Riemannian geometry of the underlying world is to be considered as the theory of these connected Euclidean tangent spaces. The generalization that we have in mind now consists of the following ... 

Rather than a Euclidean geometry we have, a non-Euclidean geometry in each tangent space, with our quadric as absolute plane in the Cayley sense. Our new geometry therefore is the overall theory of this set of Cayley spaces, in the same way that Riemannian geometry was the theory of the Euclidean tangent spaces of the underlying space. 

We now start with a generalization somewhat analogous to the transition from Euclidean to Riemannian geometry. We no longer demand that the equations (1) be integrable the tensors... 

It is remarkable that these are not dependent on

small region, one and only one curve of the system goes through two specified points. The paths are a generalization of the geodesics of Riemannian geometry. 

The theory of a general projective second rank tensor of index 2 may be considered as a generalization of the non-Euclidean geometry considered in the previous chapter. A second-rank tensor produces a metrical geometry, viz. the Riemannian geometry given by the affine tensor g j. The is dehned by the equation... 

Eisenhart, L.P. Riemannian Geometry. Princeton University Press. 

Eisenhart, L.P. Non-Riemannian Geometry. New York Amer. Math. Soc. Colloquium... 

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