Riemannian manifolds in noncommutative geometry

We present a definition of Riemannian manifold in noncommutative geometry. Using products of unbounded Kasparov modules, we show one can obtain such Riemannian manifolds from noncommutative spin^c${}^c$ manifolds; and conversely, in the presence of a spin^c${}^c$ structure. We also show how to obtain an analogue of Kasparov’s fundamental class for a Riemannian manifold, and the associated notion of Poincaré duality. Along the way we clarify the bimodule and first-order conditions for spectral triples.     

Noncommutative Riemannian geometry on graphs

We show that arising out of noncommutative geometry is a natural family of edge Laplacians on the edges of a graph. The family includes a canonical edge Laplacian associated to the graph, extending the usual graph Laplacian on vertices, and we find its spectrum. We show that for a connected graph its eigenvalues are strictly positive aside from one mandatory zero mode, and include all the vertex degrees. Our edge Laplacian is not the graph Laplacian on the line graph but rather it arises as the noncommutative Laplace-Beltrami operator on differential 1-forms, where we use the language of differential algebras to functorially interpret a graph as providing a ‘finite manifold structure’ on the set of vertices. We equip any graph with a canonical ‘Euclidean metric’ and a canonical bimodule connection, and in the case of a Cayley graph we construct a metric compatible connection for the Euclidean metric. We make use of results on bimodule connections on inner calculi on algebras, which we prove, including a general relation between zero curvature and the braid relations.     

Wave function of a free electron in a laser plasma via Riemannian geometry

The wave function of a free electron in a laser plasma described via Riemannian geometry is derived by solving the Dirac equation in the associated curved space-time. If the laser field vanishes, the wave function naturally reduces to the case in flat space-time.     

Riemannian geometry of irrotational vortex acoustics

We consider acoustic propagation in an irrotational vortex, using the technical machinery of differential geometry to investigate the "acoustic geometry" that is probed by the sound waves. The acoustic space-time curvature of a constant circulation hydrodynamical vortex leads to deflection of phonons at appreciable distances from the vortex core. The scattering angle for phonon rays is shown to be quadratic in the small quantity Gamma/2pi(cb), where Gamma is the vortex circulation, c the speed of sound, and b the impact parameter.     

Quantum and braided group Riemannian geometry

We formulate quantum group Riemannian geometry as a gauge theory of quantum differential forms. We first develop (and slightly generalise) classical Riemannian geometry in a self-dual manner as a principal bundle frame resolution and a dual pair of canonical forms. The role of Levi-Civita connection is naturally generalised to connections with vanishing torsion and cotorsion, which we introduce. We then provide the corresponding quantum group and braided group formulations with the universal quantum differential calculus. We also give general constructions, for example, including quantum spheres and quantum planes.     

Riemannian geometry in spaces with Grassman coordinates

The use of spaces containing Grassman (anticommuting) coordinates (in addition to the usual space-time coordinates) as a framework for unified gauge theories is described. The theory developed represents a local gauge-invariant extension of conventional (global) supersymmetry. Aside from containing the usual general coordinate invariance group of gravitational theory, the gauge supersymmetry group is seen to also encompass other symmetries of particle physics, e.g., electromagnetic (or Yang-Mills) invariance. The role of spontaneous symmetry breaking and the field equations unifying the Einstein, Maxwell, and Dirac interactions are discussed.Research supported in part by the National Science Foundation.Invited talk at the conference, The Riddle of Gravitation, on the Occasion of the 60th Birthday of Peter G. Bergmann, Syracuse, New York, March 1975.     

A Riemannian geometry for thermodynamic state space

We consider a class of thermodynamic systems in which the dynamics of the spontaneous approach to equilibrium is governed by the gradient of negentropy, where the gradient is taken with respect to a Riemannian metric. In open systems (dissipative structures) this gradient field is superposed with a vector field of interactions with environment. We consider three characteristics of the economy of dissipative structures: negentropy inflow (income), negentropy consumption (i.e. entropy production), and negentropy surplus (reserves). We derive explicit formulas for these characteristics and for the relations between them.     

The Riemannian geometry of the Yang-Mills moduli space

The moduli space of self-dual connections over a Riemannian 4-manifold has a natural Riemannian metric, inherited from theL ² metric on the space of connections. We give a formula for the curvature of this metric in terms of the relevant Green operators. We then examine in great detail the moduli space ₁ ofk=1 instantons on the 4-sphere, and obtain an explicit formula for the metric in this case. In particular, we prove that ₁ is rotationally symmetric and has finite geometry: it is an incomplete 5-manifold with finite diameter and finite volume.Partially supported by Horace Rackham Faculty Research Grant from the University of MichiganPartially supported by N.S.F. Grant DMS-8603461     

Geometric calculus: A new computational tool for Riemannian geometry

We compare geometric calculus applied to Riemannian geometry with Cartan's exterior calculus method. The correspondence between the two methods is clearly established. The results obtained by a package written in an algebraic language and doing general manipulations on multivectors are compared. We see that the geometric calculus is as powerful as exterior calculus.     

The spectral geometry of the canonical Riemannian submersion of a compact Lie group

Let G$G$ be a compact connected Lie group which is equipped with a bi-invariant Riemannian metric. Let m(x,y)=xy$m(x,y)=xy$ be the multiplication operator. We show the associated fibration m:G×G→G$m:G\times G\to G$ is a Riemannian submersion with totally geodesic fibers and we study the spectral geometry of this submersion. We show that the pull-backs of eigenforms on the base have finite Fourier series on the total space and we give examples where arbitrarily many Fourier coefficients can be non-zero. We give necessary and sufficient conditions for the pull-back of a form on the base to be harmonic on the total space.     

Minimal curvature trajectories: Riemannian geometry concepts for slow manifold computation in chemical kinetics

In dissipative ordinary differential equation systems different time scales cause anisotropic phase volume contraction along solution trajectories. Model reduction methods exploit this for simplifying chemical kinetics via a time scale separation into fast and slow modes. The aim is to approximate the system dynamics with a dimension-reduced model after eliminating the fast modes by enslaving them to the slow ones via computation of a slow attracting manifold. We present a novel method for computing approximations of such manifolds using trajectory-based optimization. We discuss Riemannian geometry concepts as a basis for suitable optimization criteria characterizing trajectories near slow attracting manifolds and thus provide insight into fundamental geometric properties of multiple time scale chemical kinetics. The optimization criteria correspond to a suitable mathematical formulation of “minimal relaxation” of chemical forces along reaction trajectories under given constraints. We present various geometrically motivated criteria and the results of their application to four test case reaction mechanisms serving as examples. We demonstrate that accurate numerical approximations of slow invariant manifolds can be obtained.     

Geometry and analysis on isospectral sets. I. Riemannian geometry,asymptotic case

A short introduction to the analytical and algebraic aspects of integrable systems is given. We consider the Riemannian geometry of the isospectral set belonging to the Dirichlet problem −y′' + q(x)y = λy, y(0) = y(1) = 0, where q is a square integrable function of the real Hilbert space L²_ℝ([0,1]). We derive the metric and the connection for the isospectral set, which is an infinite dimensional real analytic submanifold of LL²_ℝ([0,1 ]), in the case of large eigenvalues. The curvature in the asymptotic case is then derived and it is proved that the connection and the curvature are well defined if we take their coefficients in the discrete Sobolev spaces. We further give the explicit formulae for the parallel transport and a sufficiency condition is derived such that a curve on the isospectral set is a geodesic.     

Riemannian quantum circuit

Number theory is an abstract mathematical field that has found a fertile environment for development in theoretical physics. In particular, several physical systems were related to the zeros of the Riemann-zeta function. In this work we present the theory of a unitary matrix related to a finite number of zeros of the Riemann-zeta function. The equivalent quantum circuit and the calculation of the entanglement of a multipartite quantum state produced by the Riemannian quantum circuit are also shown.     

Fs/ns-dual-pulse orthogonal geometry plasma plume reheating for copper-based-alloys analysis

Plasma plume emission spectroscopy signal enhancements between 12- and 280-fold were obtained in air at atmospheric pressure by reheating the fs-laser ablation plume (energy 0.75 and 3.0 mJ) with a 45 mJ ns-pulse in orthogonal geometry. The emission enhancements induced by the double pulse configuration (DP) at various inter-pulse delay times and distances of the second laser beam from the target surface were investigated for copper-based-alloy standards. Temporal surveys of the plasma plume temperatures induced by both fs-single pulse (fs-SP) and DP placed at a fixed distance of 0.5 mm from the target surface were carried out. Several copper-based-alloy standards were employed for drawing Zn calibration curves by using either fs-SP or DP configurations and considering Cu as internal standard. The experimental data show that, for high Zn contents, the fs-SP set-up is affected by a self-absorption phenomenon so that a deviation from the assumed calibration single linear response is observed and two linear regressions are considered. Conversely, it has been observed that the DP configuration is not affected by any self-absorption effect and provides an improvement of the Zn limit of detection (LOD) but worse calibration linear regressions than the fs-SP. Thus, the DP scheme can increase the analytical sensitivity of fs-SP and, furthermore, its process can be supposed to be independent from the matrix composition even for largely different Zn contents of the Cu-based-alloy standards used.     

The influence of dust particle geometry on its charge and plasma potential

In this paper, a self‐consistent numerical model describing the behaviour of plasma around isolated highly charged dust particles with different shapes of rotation figures is presented. Dust particles in the form of a sphere, oblate ellipsoids (disk‐like particles), and elongated ellipsoids (rod‐like particles) are considered in the presence of an external electric field. Using the developed model, self‐consistent distributions of a space charge and plasma potential are obtained around non‐spherical dust particles. These distributions are carefully analysed by decomposing them in a series of Legendre polynomials. Decompositions of these distributions are compared with particles of different geometry. In addition, for different geometries of dust particles, the dependencies of the charge of a dust particle on geometry in the absence of an external field are investigated.