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Discrétisation de zeta-déterminants d’opérateurs de Schrödinger sur le tore

Laurent Chaumard (2006)

Bulletin de la Société Mathématique de France

Nous donnons ici deux résultats sur le déterminant ζ -régularisé det ζ A d’un opérateur de Schrödinger A = Δ g + V sur une variété compacte ℳ . Nous construisons, pour ℳ = S 1 × S 1 , une suite ( G n , ρ n , Δ n ) où G n est un graphe fini qui se plonge dans ℳ via ρ n de telle manière que ρ n ( G n ) soit une triangulation de ℳ et où  Δ n est un laplacien discret sur G n tel que pour tout potentiel V sur ℳ , la suite de réels det ( Δ n + V ) converge après renormalisation vers det ζ ( Δ g + V ) . Enfin, nous donnons sur toute variété riemannienne compacte ( ℳ , g ) de dimension inférieure ou égale à 3 ...

Distinguished connections on ( J 2 = ± 1 ) -metric manifolds

Fernando Etayo, Rafael Santamaría (2016)

Archivum Mathematicum

We study several linear connections (the first canonical, the Chern, the well adapted, the Levi Civita, the Kobayashi-Nomizu, the Yano, the Bismut and those with totally skew-symmetric torsion) which can be defined on the four geometric types of ( J 2 = ± 1 ) -metric manifolds. We characterize when such a connection is adapted to the structure, and obtain a lot of results about coincidence among connections. We prove that the first canonical and the well adapted connections define a one-parameter family of adapted...

Distinguished geodesics and jacobi fields on first order jet spaces

Vladimir Balan, Nicoleta Voicu (2004)

Open Mathematics

In the framework of jet spaces endowed with a non-linear connection, the special curves of these spaces (h-paths, v-paths, stationary curves and geodesics) which extend the corresponding notions from Riemannian geometry are characterized. The main geometric objects and the paths are described and, in the case when the vertical metric is independent of fiber coordinates, the first two variations of energy and the extended Jacobi field equations are derived.

Distinguished Riemann-Hamilton geometry in the polymomentum electrodynamics

Alexandru Oană, Mircea Neagu (2012)

Communications in Mathematics

In this paper we develop the distinguished (d-) Riemannian differential geometry (in the sense of d-connections, d-torsions, d-curvatures and some geometrical Maxwell-like and Einstein-like equations) for the polymomentum Hamiltonian which governs the multi-time electrodynamics.

Divergence operators and odd Poisson brackets

Yvette Kosmann-Schwarzbach, Juan Monterde (2002)

Annales de l’institut Fourier

We define the divergence operators on a graded algebra, and we show that, given an odd Poisson bracket on the algebra, the operator that maps an element to the divergence of the hamiltonian derivation that it defines is a generator of the bracket. This is the “odd laplacian”, Δ , of Batalin-Vilkovisky quantization. We then study the generators of odd Poisson brackets on supermanifolds, where divergences of graded vector fields can be defined either in terms of berezinian volumes or of graded connections. Examples...

Do Barbero-Immirzi connections exist in different dimensions and signatures?

L. Fatibene, Mauro Francaviglia, S. Garruto (2012)

Communications in Mathematics

We shall show that no reductive splitting of the spin group exists in dimension 3 ≤ m ≤ 20 other than in dimension m = 4 . In dimension 4 there are reductive splittings in any signature. Euclidean and Lorentzian signatures are reviewed in particular and signature ( 2 , 2 ) is investigated explicitly in detail. Reductive splittings allow to define a global SU ( 2 ) -connection over spacetime which encodes in an weird way the holonomy of the standard spin connection. The standard Barbero-Immirzi (BI) connection used in LQG is...

Dolbeault homotopy theory and compact nilmanifolds

L. Cordero, M. Fernández, A. Gray, L. Ugarte (1998)

Banach Center Publications

In this paper we study the degeneration of both the cohomology and the cohomotopy Frölicher spectral sequences in a special class of complex manifolds, namely the class of compact nilmanifolds endowed with a nilpotent complex structure. Whereas the cohomotopy spectral sequence is always degenerate for such a manifold, there exist many nilpotent complex structures on compact nilmanifolds for which the classical Frölicher spectral sequence does not collapse even at the second term.

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