UNIT VECTOR FIELDS ON ANTIPODALLY PUNCTURED SPHERES: BIG INDEX, BIG VOLUME

We establish in this paper a lower bound for the volume of a unit vector field (v) over right arrow defined ou S(n) \ {+/-x}, n = 2,3. This lower bound is related to the sum of the absolute values of the indices of (v) over right arrow at x and -x.

Maslov index in semi-Riemannian submersions

We study focal points and Maslov index of a horizontal geodesic gamma : I -> M in the total space of a semi-Riemannian submersion pi : M -> B by determining an explicit relation with the corresponding objects along the projected geodesic pi omicron gamma : I -> B in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.

Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Given a Lorentzian manifold (M,g), a geodesic gamma in M and a timelike Jacobi field Y along gamma, we introduce a special class of instants along gamma that we call Y-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Y-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Y is obtained as the restriction of a globally defined timelike Killing vector field.

COMPARISON RESULTS FOR CONJUGATE AND FOCAL POINTS IN SEMI-RIEMANNIAN GEOMETRY VIA MASLOV INDEX

We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.

Polymorphisms in genes involved in neurodevelopment may be associated with altered brain morphology in schizophrenia: Preliminary evidence

An abnormality in neurodevelopment is one of the most robust etiologic hypotheses in schizophrenia (SZ). There is also strong evidence that genetic factors may influence abnormal neurodevelopment in the disease. The present study evaluated in SZ patients, whose brain structural data had been obtained with magnetic resonance imaging (MRI), the possible association between structural brain measures, and 32 DNA polymorphisms,located in 30 genes related to neurogenesis and brain development. DNA was extracted from peripheral blood cells of 25 patients with schizophrenia, genotyping was performed using diverse procedures, and putative associations were evaluated by standard statistical methods (using the software Statistical Package for Social Sciences - SPSS) with a modified Bonferroni adjustment. For reelin (RELN), a protease that guides neurons in the developing brain and underlies neurotransmission and synaptic plasticity in adults, an association was found for a non-synonymous polymorphism (Va1997Leu) with left and right ventricular enlargement. A putative association was also found between protocadherin 12 (PCDH12), a cell adhesion molecule involved in axonal guidance and synaptic specificity, and cortical folding (asymmetry coefficient of gyrification index). Although our results are preliminary, due to the small number of individuals analyzed, such an approach could reveal new candidate genes implicated in anomalous neurodevelopment in schizophrenia. (c) 2007 Elsevier Ireland Ltd. All rights reserved.