Sub-Riemannian geodesics on nested principal bundles

We study the interplay between geodesics on two non-holono\-mic systems that are related by the action of a Lie group on them. After some geometric preliminaries, we use the Hamiltonian formalism to write the parametric form of geodesics. We present several geometric examples, including a non-holonomic structure on the Gromoll-Meyer exotic sphere and twistor space.Comment: 10 page

On the Alexandrov Topology of sub-Lorentzian Manifolds

It is commonly known that in Riemannian and sub-Riemannian Geometry, the metric tensor on a manifold defines a distance function. In Lorentzian Geometry, instead of a distance function it provides causal relations and the Lorentzian time-separation function. Both lead to the definition of the Alexandrov topology, which is linked to the property of strong causality of a space-time. We studied three possible ways to define the Alexandrov topology on sub-Lorentzian manifolds, which usually give different topologies, but agree in the Lorentzian case. We investigated their relationships to each other and the manifold's original topology and their link to causality.Comment: 20 page

Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle

In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.Comment: 40 pages. Corrected typos and improved redactio

Conformal loop ensembles and the stress-energy tensor

We give a construction of the stress-energy tensor of conformal field theory (CFT) as a local "object" in conformal loop ensembles CLE_\kappa, for all values of \kappa in the dilute regime 8/3 < \kappa <= 4 (corresponding to the central charges 0 < c <= 1, and including all CFT minimal models). We provide a quick introduction to CLE, a mathematical theory for random loops in simply connected domains with properties of conformal invariance, developed by Sheffield and Werner (2006). We consider its extension to more general regions of definition, and make various hypotheses that are needed for our construction and expected to hold for CLE in the dilute regime. Using this, we identify the stress-energy tensor in the context of CLE. This is done by deriving its associated conformal Ward identities for single insertions in CLE probability functions, along with the appropriate boundary conditions on simply connected domains; its properties under conformal maps, involving the Schwarzian derivative; and its one-point average in terms of the "relative partition function." Part of the construction is in the same spirit as, but widely generalizes, that found in the context of SLE_{8/3} by the author, Riva and Cardy (2006), which only dealt with the case of zero central charge in simply connected hyperbolic regions. We do not use the explicit construction of the CLE probability measure, but only its defining and expected general properties.Comment: 49 pages, 3 figures. This is a concatenated, reduced and simplified version of arXiv:0903.0372 and (especially) arXiv:0908.151

The rolling problem: overview and challenges

In the present paper we give a historical account -ranging from classical to modern results- of the problem of rolling two Riemannian manifolds one on the other, with the restrictions that they cannot instantaneously slip or spin one with respect to the other. On the way we show how this problem has profited from the development of intrinsic Riemannian geometry, from geometric control theory and sub-Riemannian geometry. We also mention how other areas -such as robotics and interpolation theory- have employed the rolling model.Comment: 20 page

Direct Observations of Sigma Phase Formation in Duplex Stainless Steels using In Situ Synchrotron X-Ray Diffraction

The formation and growth of sigma phase in 2205 duplex stainless steel was observed and measured in real time using synchrotron radiation during 10 hr isothermal heat treatments at temperatures between 700 C and 850 C. Sigma formed in near-equilibrium quantities during the isothermal holds, starting from a microstructure which contained a balanced mixture of metastable ferrite and austenite. In situ synchrotron diffraction continuously monitored the transformation, and these results were compared to those predicted by thermodynamic calculations. Differences between the calculated and measured amounts of sigma, ferrite and austenite suggest that the thermodynamic calculations underpredict the sigma dissolution temperature by approximately 50 C. The data were further analyzed using a modified Johnson-Mehl-Avrami (JMA) approach to determine kinetic parameters for sigma formation over this temperature range. The initial JMA exponent, n, at low fractions of sigma was found to be approximately 7.0, however, towards the end of the transformation, n decreased to values of approximately 0.75. The change in the JMA exponent was attributed to a change in the transformation mechanism from discontinuous precipitation with increasing nucleation rate, to growth of the existing sigma phase after nucleation site saturation occurred. Because of this change in mechanism, it was not possible to determine reliable values for the activation energy and pre-exponential terms for the JMA equation. While cooling back to room temperature, the partial transformation of austenite resulted in a substantial increase in the ferrite content, but sigma retained its high temperature value to room temperature