Bi-Legendrian manifolds and paracontact geometry

We study the interplays between paracontact geometry and the theory of bi-Legendrian manifolds. We interpret the bi-Legendrian connection of a bi-Legendrian manifold M as the paracontact connection of a canonical paracontact structure induced on M and then we discuss many consequences of this result both for bi-Legendrian and for paracontact manifolds. Finally new classes of examples of paracontact manifolds are presented.Comment: to appear in Int. J. Geom. Meth. Mod. Phy

Convex Functions and Spacetime Geometry

Convexity and convex functions play an important role in theoretical physics. To initiate a study of the possible uses of convex functions in General Relativity, we discuss the consequences of a spacetime $(M,g_{\mu \nu})$ or an initial data set $(\Sigma, h_{ij}, K_{ij})$ admitting a suitably defined convex function. We show how the existence of a convex function on a spacetime places restrictions on the properties of the spacetime geometry.Comment: 26 pages, latex, 7 figures, improved version. some claims removed, references adde

The geometry of nonlinear least squares with applications to sloppy models and optimization

Parameter estimation by nonlinear least squares minimization is a common problem with an elegant geometric interpretation: the possible parameter values of a model induce a manifold in the space of data predictions. The minimization problem is then to find the point on the manifold closest to the data. We show that the model manifolds of a large class of models, known as sloppy models, have many universal features; they are characterized by a geometric series of widths, extrinsic curvatures, and parameter-effects curvatures. A number of common difficulties in optimizing least squares problems are due to this common structure. First, algorithms tend to run into the boundaries of the model manifold, causing parameters to diverge or become unphysical. We introduce the model graph as an extension of the model manifold to remedy this problem. We argue that appropriate priors can remove the boundaries and improve convergence rates. We show that typical fits will have many evaporated parameters. Second, bare model parameters are usually ill-suited to describing model behavior; cost contours in parameter space tend to form hierarchies of plateaus and canyons. Geometrically, we understand this inconvenient parametrization as an extremely skewed coordinate basis and show that it induces a large parameter-effects curvature on the manifold. Using coordinates based on geodesic motion, these narrow canyons are transformed in many cases into a single quadratic, isotropic basin. We interpret the modified Gauss-Newton and Levenberg-Marquardt fitting algorithms as an Euler approximation to geodesic motion in these natural coordinates on the model manifold and the model graph respectively. By adding a geodesic acceleration adjustment to these algorithms, we alleviate the difficulties from parameter-effects curvature, improving both efficiency and success rates at finding good fits.Comment: 40 pages, 29 Figure

3-quasi-Sasakian manifolds

In the present paper we carry on a systematic study of 3-quasi-Sasakian manifolds. In particular we prove that the three Reeb vector fields generate an involutive distribution determining a canonical totally geodesic and Riemannian foliation. Locally, the leaves of this foliation turn out to be Lie groups: either the orthogonal group or an abelian one. We show that 3-quasi-Sasakian manifolds have a well-defined rank, obtaining a rank-based classification. Furthermore, we prove a splitting theorem for these manifolds assuming the integrability of one of the almost product structures. Finally, we show that the vertical distribution is a minimum of the corrected energy.Comment: 17 pages, minor modifications, references update

Geometry and field theory in multi-fractional spacetime

We construct a theory of fields living on continuous geometries with fractional Hausdorff and spectral dimensions, focussing on a flat background analogous to Minkowski spacetime. After reviewing the properties of fractional spaces with fixed dimension, presented in a companion paper, we generalize to a multi-fractional scenario inspired by multi-fractal geometry, where the dimension changes with the scale. This is related to the renormalization group properties of fractional field theories, illustrated by the example of a scalar field. Depending on the symmetries of the Lagrangian, one can define two models. In one of them, the effective dimension flows from 2 in the ultraviolet (UV) and geometry constrains the infrared limit to be four-dimensional. At the UV critical value, the model is rendered power-counting renormalizable. However, this is not the most fundamental regime. Compelling arguments of fractal geometry require an extension of the fractional action measure to complex order. In doing so, we obtain a hierarchy of scales characterizing different geometric regimes. At very small scales, discrete symmetries emerge and the notion of a continuous spacetime begins to blur, until one reaches a fundamental scale and an ultra-microscopic fractal structure. This fine hierarchy of geometries has implications for non-commutative theories and discrete quantum gravity. In the latter case, the present model can be viewed as a top-down realization of a quantum-discrete to classical-continuum transition.Comment: 1+82 pages, 1 figure, 2 tables. v2-3: discussions clarified and improved (especially section 4.5), typos corrected, references added; v4: further typos correcte

Exponential Barycenters of the Canonical Cartan Connection and Invariant Means on Lie Groups

International audienceWhen performing statistics on elements of sets that possess a particular geometric structure, it is desirable to respect this structure. For instance in a Lie group, it would be judicious to have a notion of a mean which is stable by the group operations (composition and inversion). Such a property is ensured for Riemannian center of mass in Lie groups endowed with a bi-invariant Riemannian metric, like compact Lie groups (e.g. rotations). However, bi-invariant Riemannian metrics do not exist for most non compact and non-commutative Lie groups. This is the case in particular for rigid-body transformations in any dimension greater than one, which form the most simple Lie group involved in biomedical image registration. In this paper, we propose to replace the Riemannian metric by an affine connection structure on the group. We show that the canonical Cartan connections of a connected Lie group provides group geodesics which are completely consistent with the composition and inversion. With such a non-metric structure, the mean cannot be defined by minimizing the variance as in Riemannian Manifolds. However, the characterization of the mean as an exponential barycenter gives us an implicit definition of the mean using a general barycentric equation. Thanks to the properties of the canonical Cartan connection, this mean is naturally bi-invariant. We show the local existence and uniqueness of the invariant mean when the dispersion of the data is small enough. We also propose an iterative fixed point algorithm and demonstrate that the convergence to the invariant mean is at least linear. In the case of rigid-body transformations, we give a simple criterion for the global existence and uniqueness of the bi-invariant mean, which happens to be the same as for rotations. We also give closed forms for the bi-invariant mean in a number of simple but instructive cases, including 2D rigid transformations. For general linear transformations, we show that the bi-invariant mean is a generalization of the (scalar) geometric mean, since the determinant of the bi-invariant mean is the geometric mean of the determinants of the data. Finally, we extend the theory to higher order moments, in particular with the covariance which can be used to define a local bi-invariant Mahalanobis distance