Holonomy and submanifold geometry

We survey applications of holonomic methods to the study of submanifold geometry, showing the consequences of some sort of extrinsic version of de Rham decomposition and Berger's Theorem, the so-called Normal Holonomy Theorem. At the same time, from geometric methods in submanifold theory we sketch very strong applications to the holonomy of Lorentzian manifolds. Moreover we give a conceptual modern proof of a result of Kostant for homogeneous space

Critical behavior of plastic depinning of vortex lattices in two dimensions: Molecular dynamics simulations

Using molecular dynamics simulations, we report a study of the dynamics of two-dimensional vortex lattices driven over a disordered medium. In strong disorder, when topological order is lost, we show that the depinning transition is analogous to a second order critical transition: the velocity-force response at the onset of motion is continuous and characterized by critical exponents. Combining studies at zero and nonzero temperature and using a scaling analysis, two critical expo- nents are evaluated. We find v\sim (F-F_c)^\beta with \beta=1.3\pm0.1 at T=0 and F>F_c, and v\sim T^{1/\delta} with \delta^{-1}=0.75\pm0.1 at F=F_c, where F_c is the critical driving force at which the lattice goes from a pinned state to a sliding one. Both critical exponents and the scaling function are found to exhibit universality with regard to the pinning strength and different disorder realizations. Furthermore, the dynamics is shown to be chaotic in the whole critical region.Comment: 8 pages, 6 figure

The elastic depinning transition of vortex lattices in two dimensions

Large scale numerical simulations are used to study the elastic dynamics of two-dimensional vortex lattices driven on a disordered medium in the case of weak disorder. We investigate the so-called elastic depinning transition by decreasing the driving force from the elastic dynamical regime to the state pinned by the quenched disorder. Similarly to the plastic depinning transition, we find results compatible with a second order phase transition, although both depinning transitions are very different from many viewpoints. We evaluate three critical exponents of the elastic depinning transition. $\beta = 0.29 \pm 0.03$ is found for the velocity exponent at zero temperature, and from the velocity-temperature curves we extract the critical exponent $\delta^{-1} = 0.28 \pm 0.05$. Furthermore, in contrast with charge-density waves, a finite-size scaling analysis suggests the existence of a unique diverging length at the depinning threshold with an exponent $\nu= 1.04 \pm 0.04$, which controls the critical force distribution, the finite-size crossover force distribution and the intrinsic correlation length. Finally, a scaling relation is found between velocity and temperature with the $\beta$ and $\delta$ critical exponents both independent with regard to pinning strength and disorder realizations.Comment: 17 pages, 10 figure

Surfaces in R4 with constant principal angles with respect to a plane

We study surfaces in R4 whose tangent spaces have constant principal angles with respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of $\R^2$. We classify all surfaces with one principal angle equal to $0$ and observe that they can be constructed as the union of normal holonomy tubes. We also classify the complete constant angles surfaces in R4 with respect to a plane. They turn out to be extrinsic products. We characterize which surfaces with constant principal angles are compositions in the sense of Dajczer-Do Carmo. Finally, we classify surfaces with constant principal angles contained in a sphere and those with parallel mean curvature vector fiel

HOMOGENEOUS RIEMANNIAN MANIFOLDS WITH NON-TRIVIAL NULLITY

We develop a general theory for irreducible homogeneous spaces M = G/H, in relation to the nullity distribution ν of their curvature tensor. We construct natural invariant (different and increasing) distributions associated with the nullity, that give a deep insight of such spaces. In particular, there must exist an order-two transvection, not in the nullity, with null Jacobi operator. This fact was very important for finding out the first homogeneous examples with non-trivial nullity, i.e., where the nullity distribution is not parallel. Moreover, we construct irreducible examples of conullity k = 3, the smallest possible, in any dimension. None of our examples admit a quotient of finite volume. We also proved that H is trivial and G is solvable if k = 3. Another of our main results is that the leaves, i.e., the integral manifolds, of the nullity are closed (we used a rather delicate argument). This implies that M is a Euclidean affine bundle over the quotient by the leaves of ν. Moreover, we prove that ν⊥ defines a metric connection on this bundle with transitive holonomy or, equivalently, ν⊥ is completely non-integrable (this is not in general true for an arbitrary autoparallel and at invariant distribution). We also found some general obstruction for the existence of non-trivial nullity: e.g., if G is reductive (in particular, if M is compact), or if G is two-step nilpotent

Holonomy and submanifold geometry

We survey applications of holonomic methods to the study of submanifold geometry, showing the consequences of some sort of extrinsic version of de Rham decomposition and Berger's Theorem, the so-called Normal Holonomy Theorem. At the same time, from geometric methods in submanifold theory we sketch very strong applications to the holonomy of Lorentzian manifolds. Moreover we give a conceptual modern proof of a result of Kostant for homogeneous spaces

Efficacy of a partially hydrolyzed whey formula on infant colic: a randomized controlled trial

Background: Infant colic (IC) affects up to 20% of infants in the first 4 months of life. Although IC is a benign affection that spontaneously resolves after the first 3-4 months of life, it is often a stressful problem for parents. Methods: Babies, aged ≤ 3 months, observed at family pediatrician office because a suspect of IC, were randomized in two groups of 3-week dietary intervention: Group 1, receiving non-analgesic, non-nutritive soothing maneuvers, continuing a standard formula; Group 2, receiving a partially hydrolyzed whey formula (w-pHF), containing GOS (0.5g/100ml), low content of lactose (2.5g/100ml) and low osmolarity (185 mOsm). All infants performed clinical examinations at enrollment and after 7, 14 and 21 days. Number of colic episodes, and the number and consistency of fecal outputs were recorded daily. Results: Fifty infants with IC were enrolled and randomized: 25 in Group 1 and 25 in Group 2. The rate of infants with IC in Group 2 decreased significantly within 14 days compared to Group 1 and the number of bowel movements increased significantly within 7 days in Group 2 compared to Group 1. Stool consistency significantly improved in Group 2 within 7 days. Conclusion: The studied formula could represent a useful approach in infants with IC reducing pharmacological treatments