Metric connections in projective differential geometry

We search for Riemannian metrics whose Levi-Civita connection belongs to a given projective class. Following Sinjukov and Mikes, we show that such metrics correspond precisely to suitably positive solutions of a certain projectively invariant finite-type linear system of partial differential equations. Prolonging this system, we may reformulate these equations as defining covariant constant sections of a certain vector bundle with connection. This vector bundle and its connection are derived from the Cartan connection of the underlying projective structure.Comment: 10 page

Further functional determinants

Functional determinants for the scalar Laplacian on spherical caps and slices, flat balls, shells and generalised cylinders are evaluated in two, three and four dimensions using conformal techniques. Both Dirichlet and Robin boundary conditions are allowed for. Some effects of non-smooth boundaries are discussed; in particular the 3-hemiball and the 3-hemishell are considered. The edge and vertex contributions to the $C_{3/2}$ coefficient are examined.Comment: 25 p,JyTex,5 figs. on request

Positive mass theorem for the Paneitz-Branson operator

We prove that under suitable assumptions, the constant term in the Green function of the Paneitz-Branson operator on a compact Riemannian manifold $(M,g)$ is positive unless $(M,g)$ is conformally diffeomophic to the standard sphere. The proof is inspired by the positive mass theorem on spin manifolds by Ammann-Humbert.Comment: 7 page

Prolongations of Geometric Overdetermined Systems

We show that a wide class of geometrically defined overdetermined semilinear partial differential equations may be explicitly prolonged to obtain closed systems. As a consequence, in the case of linear equations we extract sharp bounds on the dimension of the solution space.Comment: 22 pages. In the second version, a comparison with the classical theory of prolongations was added. In this third version more details were added concerning our construction and especially the use of Kostant's computation of Lie algebra cohomolog

The C_2 heat-kernel coefficient in the presence of boundary discontinuities

We consider the heat-kernel on a manifold whose boundary is piecewise smooth. The set of independent geometrical quantities required to construct an expression for the contribution of the boundary discontinuities to the C_{2} heat-kernel coefficient is derived in the case of a scalar field with Dirichlet and Robin boundary conditions. The coefficient is then determined using conformal symmetry and evaluation on some specific manifolds. For the Robin case a perturbation technique is also developed and employed. The contributions to the smeared heat-kernel coefficient and cocycle function are calculated. Some incomplete results for spinor fields with mixed conditions are also presented.Comment: 25 pages, LaTe

Tract-Based Spatial Statistics in Preterm-Born Neonates Predicts Cognitive and Motor Outcomes at 18 Months.

BACKGROUND AND PURPOSE: Adverse neurodevelopmental outcome is common in children born preterm. Early sensitive predictors of neurodevelopmental outcome such as MR imaging are needed. Tract-based spatial statistics, a diffusion MR imaging analysis method, performed at term-equivalent age (40 weeks) is a promising predictor of neurodevelopmental outcomes in children born very preterm. We sought to determine the association of tract-based spatial statistics findings before term-equivalent age with neurodevelopmental outcome at 18-months corrected age. MATERIALS AND METHODS: Of 180 neonates (born at 24-32-weeks\u27 gestation) enrolled, 153 had DTI acquired early at 32 weeks\u27 postmenstrual age and 105 had DTI acquired later at 39.6 weeks\u27 postmenstrual age. Voxelwise statistics were calculated by performing tract-based spatial statistics on DTI that was aligned to age-appropriate templates. At 18-month corrected age, 166 neonates underwent neurodevelopmental assessment by using the Bayley Scales of Infant Development, 3rd ed, and the Peabody Developmental Motor Scales, 2nd ed. RESULTS: Tract-based spatial statistics analysis applied to early-acquired scans (postmenstrual age of 30-33 weeks) indicated a limited significant positive association between motor skills and axial diffusivity and radial diffusivity values in the corpus callosum, internal and external/extreme capsules, and midbrain (P \u3c .05, corrected). In contrast, for term scans (postmenstrual age of 37-41 weeks), tract-based spatial statistics analysis showed a significant relationship between both motor and cognitive scores with fractional anisotropy in the corpus callosum and corticospinal tracts (P \u3c .05, corrected). Tract-based spatial statistics in a limited subset of neonates (n = 22) scanned at CONCLUSIONS: The strength of the association between fractional anisotropy values and neurodevelopmental outcome scores increased from early-to-late-acquired scans in preterm-born neonates, consistent with brain dysmaturation in this population

A classification of local Weyl invariants in D=8

Following a purely algebraic procedure, we provide an exhaustive classification of local Weyl-invariant scalar densities in dimension D=8.Comment: LaTeX, 19 pages, typos corrected, one reference adde

On the conformal transformation and duality in gravity

The theory described by the sum of the Einstein-Hilbert action and the action of conformal scalar field possesses the duality symmetry which includes some special conformal transformation of the metric, and also inversion of scalar field and of the gravitational constant. In the present paper the conformal duality is generalized for arbitrary space-time dimension $n \neq 2$ and for the general sigma-model type conformal scalar theory. We also consider to apply the conformal duality for the investigation of quantum gravity in the strong curvature regime. The trace of the first coefficient of the Schwinger-DeWitt expansion is derived and it's dependence on the gauge fixing condition is considered. After that we discuss the way to extract the gauge-fixing independent result and also it's possible physical applications.Comment: LaTeX, 15 pages, no figures. To appear in Classical and Quantum Gravit