Sobolev Metrics on Diffeomorphism Groups and the Derived Geometry of Spaces of Submanifolds

Given a finite dimensional manifold $N$, the group $\operatorname{Diff}_{\mathcal S}(N)$ of diffeomorphism of $N$ which fall suitably rapidly to the identity, acts on the manifold $B(M,N)$ of submanifolds on $N$ of diffeomorphism type $M$ where $M$ is a compact manifold with $\dim M<\dim N$. For a right invariant weak Riemannian metric on $\operatorname{Diff}_{\mathcal S}(N)$ induced by a quite general operator $L:\frak X_{\mathcal S}(N)\to \Gamma(T^*N\otimes\operatorname{vol}(N))$, we consider the induced weak Riemannian metric on $B(M,N)$ and we compute its geodesics and sectional curvature. For that we derive a covariant formula for curvature in finite and infinite dimensions, we show how it makes O'Neill's formula very transparent, and we use it finally to compute sectional curvature on $B(M,N)$.Comment: 28 pages. In this version some misprints correcte

A common generalization of the Fr\"olicher-Nijenhuis bracket and the Schouten bracket for symmetry multi vector fields

There is a canonical mapping from the space of sections of the bundle $\wedge T^\ast M\otimes ST\ M$ to $\Omega(T^\ast M ; T(T^\ast M))$. It is shown that this is a homomorphism on $\Omega(M;TM) for the Fr\"olicher-Nijenhuis brackets, and also on$\Gamma(ST\ M)$for the Schouten bracket of symmetric multi vector fields. But the whole image is not a subalgebra for the Fr\"olicher-Nijenhuis bracket on$\Omega(T^\ast M;T(T^\ast M))$.Comment: 14 pages, AMSTEX, LPTHE-ORSAY 94/05 and ESI 70 (1994

Intermediate valence behavior in CeCo9Si4

The novel ternary compound CeCo$_9$Si$_4$ has been studied by means of specific heat, magnetisation, and transport measurements. Single crystal X-ray Rietveld refinements reveal a fully ordered distribution of Ce, Co and Si atoms with the tetragonal space group I4/mcm isostructural with other RCo9Si4. The smaller lattice constants of CeCo9Si4 in comparison with the trend established by other RCo9Si4 is indicative for intermediate valence of cerium. While RCo9Si4 with R= Pr, .. Tb, and Y show ferromagnetism and LaCo9Si4 is nearly ferromagnetic, CeCo9Si4 remains paramagnetic even in external fields as large as 40 T, though its electronic specific heat coefficient (g~190 mJ/molK^2) is of similar magnitude as that of metamagnetic LaCo9Si4 and weakly ferromagnetic YCo9Si4.Comment: 2 pages, 3 figures, submitted to SCES 0

Completely integrable systems: a generalization

We present a slight generalization of the notion of completely integrable systems to get them being integrable by quadratures. We use this generalization to integrate dynamical systems on double Lie groups.Comment: Latex, 15 page

Construction of completely integrable systems by Poisson mappings

Pulling back sets of functions in involution by Poisson mappings and adding Casimir functions during the process allows to construct completely integrable systems. Some examples are investigated in detail.Comment: AmsTeX, 9 page

Un-reduction

This paper provides a full geometric development of a new technique called un-reduction, for dealing with dynamics and optimal control problems posed on spaces that are unwieldy for numerical implementation. The technique, which was originally concieved for an application to image dynamics, uses Lagrangian reduction by symmetry in reverse. A deeper understanding of un-reduction leads to new developments in image matching which serve to illustrate the mathematical power of the technique.Comment: 25 pages, revised versio

On the monotonicity of scalar curvature in classical and quantum information geometry

We study the statistical monotonicity of the scalar curvature for the alpha-geometries on the simplex of probability vectors. From the results obtained and from numerical data we are led to some conjectures about quantum alpha-geometries and Wigner-Yanase-Dyson information. Finally we show that this last conjecture implies the truth of the Petz conjecture about the monotonicity of the scalar curvature of the Bogoliubov-Kubo-Mori monotone metric.Comment: 20 pages, 2 .eps figures; (v2) section 2 rewritten, typos correcte

Tetrad gravity, electroweak geometry and conformal symmetry

A partly original description of gauge fields and electroweak geometry is proposed. A discussion of the breaking of conformal symmetry and the nature of the dilaton in the proposed setting indicates that such questions cannot be definitely answered in the context of electroweak geometry.Comment: 21 pages - accepted by International Journal of Geometric Methods in Modern Physics - v2: some minor changes, mostly corrections of misprint