Riemannian

Z) is a triangle. The integral in (4,75) is, in Riemann s definition, an improper integral. It can be approximated by a finite sum because the integrand is monotone, which implies, furthermore, that is bounded. I skip the details because considerations of this type are standard in the discussion of improper Riemannian integrals, Equations (4,73) and (4.75), in the notation of (4.56), imply... 

Proof. The T-action preserves the Riemannian metric and the hyper-Kahler structure... 

Now we fix a Riemannian metric g which is invariant under the T-action. The symplectic form UJ together with the Riemannian metric g gives an almost complex structure I defined by uj v,x) = g Iv,w). With this almost complex structure, we regard the tangent space Tj-X as a complex vector space. Let X = be the decomposition into the... 

The generalized Fisher theorems derived in this section are statements about the space variation of the vectors of the relative and absolute space-specific rates of growth. These vectors have a simple natural (biological, chemical, physical) interpretation They express the capacity of a species of type u to fill out space in genetic language, they are space-specific fitness functions. In addition, the covariance matrix of the vector of the relative space-specific rates of growth, gap, [Eq. (25)] is a Riemannian metric tensor that enters the expression of a Fisher information metric [Eqs. (24) and (26)]. These results may serve as a basis for solving inverse problems for reaction transport systems. 

Representations of these and other tensors in an arbitrary system of coordinates may be constructed as follows. For each contravariant rank 2 Cartesian tensor T " (such as H ) or covariant tensor S v (such as m v), we define corresponding Riemannian representations... 

For each rank 2 contravariant Riemannian tensors T (with two raised indices) we define a. K x K projection onto the hard subspace... 

In this section, we develop some useful relationships involving the determinants and inverses of projected tensors. Let S ap be the Riemannian representation of an arbitrary symmetric covariant tensor with a Cartesian representation S v We may write the Riemannian representation in block matrix form, using the indices a,b to denote blocks in which a or p mns over the soft coordinates and i,j to represent hard coordinates, as... 

Gromov, M., "Metric Structures for Riemannian and Non-Riemannian Spaces. Progress in Mathematics, 152". Birkhauser, Boston (1999). 

Kuchar, K. (1976) Kinematics of tensor fields in hyperspace, ll. Journal of Mathematical Physics. 17(5) 792—800. (Differential geometry in hyperspace is used to investigate kinematical relationships between hypersurface projections of spacetime tensor fields in a Riemannian spacetime.)... 

M. W. Evans, P. K. Anastasovski, T. E. Bearden et al., Longitudinal Modes in Vacuo of the Electromagnetic Field in Riemannian Spacetime, submitted to Optik, 2000 (in press). 

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

Since the G-action on // 1(C) is free, the slice theorem implies that the quotient space gTl((,)/G has a structure of a C°°-manifold such that the tangent space TaAtt-HO/G) at the orbit G x is isomorphic to the orthogonal complement of Vx in Txg 1( ). Hence the tangent space is the orthogonal complement of Vx IVX JVX KVX in TxX, which is invariant under I, J and K. Thus we have the induced almost hyper-complex structure. The restriction of the Riemannian metric g induces a Riemannian metric on the quotient g 1(()/G. In order to show that these define a hyper-Kahler structure, it is enough to check that the associated Kahler forms u>[, u) 2 and co z are closed by Lemma 3.32. 

Proof. The T-action preserves the Riemannian metric and the hyper-Kahler structure on the Hilbert scheme (C2) constructed in Chapter 3. The restriction gives a hyper-Kahler structure on X. ... 

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). 

The metric geometry of equilibrium thermodynamics provides an unusual prototype in the rich spectrum of possibilities of differential geometry. Just as Einstein s general relativistic theory of gravitation enriched the classical Riemann theory of curved spaces, so does its thermodynamic manifestation suggest further extensions of powerful Riemannian concepts. Theorems and tools of the differential geometer may be sharpened or extended by application to the unique Riemannian features of equilibrium chemical and phase thermodynamics. 

B. Andresen, R. S. Berry, E. Ihrig, and R Salamon. Inducing Weinhold s metric from Euclidean and Riemannian metrics. Phys. Rev. A 37, 849-51 (1988). 

In addition, several alternative formulations of thermodynamic geometry have been presented, starting from entropy-based (or other) fundamental equations (see Sections 5.4 and 5.5). From the equilibrium thermodynamics viewpoint, these alternative formulations are completely equivalent, and each could be considered a special case of the general transformations outlined in Section 11.4. Nevertheless, each alternative may suggest distinct statistical-mechanical origins, Riemannian paths, or other connotations that make it preferable for applications outside the equilibrium thermodynamics framework. 

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. 

---