Bundle Boundary Research Articles

Worm domains and Fefferman space–time singularities

Let W be a smoothly bounded worm domain in C 2 and let A = Null ( L θ ) be the set of Levi-flat points on the boundary ∂ W of W . We study the relationship between pseudohermitian geometry of the strictly pseudoconvex locus M = ∂ W ∖ A and the theory of space–time singularities associated to the Fefferman metric F θ on the total space of the canonical circle bundle S 1 → C ( M ) ⟶ π M . Given any point ( 0 , w 0 ) ∈ A , we show that every lift Γ ( φ ) ∈ C ( M ) , 0 ≤ φ − log | w 0 | 2 < π ∕ 2 , of the circle Γ w 0 : r = 2 cos [ log | w 0 | 2 − φ ] in M , runs into a curvature singularity of Fefferman’s space–time ( C ( M ) , F θ ) . We show that Σ = π − 1 ( Γ w 0 ) is a Lorentzian real surface in ( C ( M ) , F θ ) such that the immersion ι : Σ ↪ C ( M ) has a flat normal connection. Consequently, there is a natural isometric immersion j : O ( Σ ) → O ( C ( M ) , Σ ) between the total spaces of the principal bundles of Lorentzian frames O ( 1 , 1 ) → O ( Σ ) → Σ and adapted Lorentzian frames O ( 1 , 1 ) × O ( 2 ) → O ( C ( M ) , Σ ) → Σ , endowed with Schmidt metrics, descending to a map of bundle completions which maps the b -boundary of Σ into the adapted bundle boundary of C ( M ) , i.e. j ( Σ ̇ ) ⊂ ∂ adt C ( M ) .

Uniform spatial distribution of collagen fibril radii within tendon implies local activation of pC-collagen at individual fibrils

Collagen fibril cross-sectional radii show no systematic variation between the interior and the periphery of fibril bundles, indicating an effectively constant rate of collagen incorporation into fibrils throughout the bundle. Such spatially homogeneous incorporation constrains the extracellular diffusion of collagen precursors from sources at the bundle boundary to sinks at the growing fibrils. With a coarse-grained diffusion equation we determine stringent bounds, using parameters extracted from published experimental measurements of tendon development. From the lack of new fibril formation after birth, we further require that the concentration of diffusing precursors stays below the critical concentration for fibril nucleation. We find that the combination of the diffusive bound, which requires larger concentrations to ensure homogeneous fibril radii, and lack of nucleation, which requires lower concentrations, is only marginally consistent with fully processed collagen using conservative bounds. More realistic bounds may leave no consistent concentrations. Therefore, we propose that unprocessed pC-collagen diffuses from the bundle periphery followed by local C-proteinase activity and subsequent collagen incorporation at each fibril. We suggest that C-proteinase is localized within bundles, at fibril surfaces, during radial fibrillar growth. The much greater critical concentration of pC-collagen, as compared to fully processed collagen, then provides broad consistency between homogeneous fibril radii and the lack of fibril nucleation during fibril growth.

Boundary estimation of fiber bundles derived from diffusion tensor images

Diffusion tensor imaging (DTI) is a non-invasive imaging technique that allows estimating the location of white matter tracts based on the measurement of water diffusion properties. Using DTI data, the fiber bundle boundary can be determined to gain information about eloquent structures, which is of major interest for neurosurgical interventions. In this paper, a novel approach for boundary estimation is presented. DTI in combination with diverse segmentation algorithms allows estimating the position and course of fiber tracts in the human brain. For additional information about the expansion of the fiber bundle, the introduced iterative approach uses the centerline of a tracked fiber bundle between two regions of interest (ROI). After sampling along this centerline, rays are sent out radially, discrete 2D contours are calculated, and the fiber bundle boundary is estimated in a stepwise manner. For this purpose, each ray is analyzed using several criteria, including anisotropy parameters and angle parameters, to find the boundary point. The novel method for automatically calculating the boundaries has been applied to several artificially generated DTI datasets. Multiple parameters were varied: number of rays per plane, sampling rate and sampled points along the rays. For the DTI data used in the experiments, the method yielded a dice similarity coefficient (DSC) between 74.7 and 91.5%. In this paper, a novel approach to retrieve significant information about the fiber bundle boundary from DTI data is presented. The method is a contribution to gather important knowledge about high-risk structures in neurosurgical interventions.

Hodge and signature theorems for a family of manifolds with fibre bundle boundary

Let $\bar{M}$ be a manifold with boundary $Y$ which is the total space of a fibre bundle, and is defined by the vanishing of a boundary defining function, $x$. We prove $L^2$ Hodge and signature theorems for $M$ endowed with a metric of the form $dx^2 + x^{2c} h + k$, where $k$ is the lift to $Y$ of the metric on the base of the fibre bundle, $h$ is a two form on $Y$ which restricts to a metric on each fibre, and $0 \leq c \leq 1$. These metrics interpolate between the case when $c=0$, in which case the metric near the boundary is a cylinder, and the case where $c=1$, in which case the metric near the boundary is that of a cone bundle over the base of the boundary fibration. We show that the $L^2$ Hodge theorems for the cohomologies given by the maximal and minimal extensions of $d$ with respect to these metrics and the $L^2$ signature theorem for the image of the minimal cohomology in the maximal cohomology interpolate between known results for $L^2$ Hodge and signature theorems for cylindrical and cone bundle type metrics. In particular, the Hodge theorems all relate the related spaces of $L^2$ harmonic forms to intersection cohomology of varying perversities for $X$, the space formed from $\bar{M}$ by collapsing the fibres of $Y$ at the boundary. The signature theorem involves variations on the $\tau$ invariant described by Dai.

The b-completion of the Friedmann space

We study the b-completion of the three Friedmann models of the Universe, having as models for 3-space the sphere, the Euclidean space or the hyperbolic space. We show that in the first case there is just one singularity, having the full completion as only neighborhood. In the other two cases there is one essential singularity, which is the limit of all past causal geodesics; again, it has a single neighborhood. This extends results by Bosshard [On the b-boundary of the closed Friendmann Model, Commun. Math. Phys. 46 (1976) 263–268] and Johnson [The bundle boundary in some special cases, J. Math. Phys. 18 (5) (1977) 898–902] on the closed Friedmann model.

The bundle boundary for the Schwarzschild and Friedmann solutions

The bundle boundary for the Schwarzschild and Friedmann solutions

The bundle boundary in some special cases

We examine a class of two-dimensional Lorentz manifolds which are ’’singular’’ in a certain sense. It is shown that, for such a manifold (M, g), the bundle boundary is a single point whose only neighborhood is all of M̄ [the bundle completion of M; see B. G. Schmidt, Gen. Rel. Gravitation 1, 269–80 (1971)]. The four-dimensional Schwarzschild and Friedmann–Robertson–Walker solutions are then investigated. We show that the bundle completions of these spaces are not Hausdorff.