Dirac eigenvalues and total scalar curvature

It has recently been conjectured that the eigenvalues $\lambda$ of the Dirac operator on a closed Riemannian spin manifold $M$ of dimension $n\ge 3$ can be estimated from below by the total scalar curvature: $$\lambda^2 \ge \frac{n}{4(n-1)} \cdot \frac{\int_M S}{vol(M)}.$$ We show by example that such an estimate is impossible.Comment: 9 pages, LaTeX, uses pstricks macro package. to appear in Journal of Geometry and Physic

Path integrals on manifolds by finite dimensional approximation

Let M be a compact Riemannian manifold without boundary and let H be a self-adjoint generalized Laplace operator acting on sections in a bundle over M. We give a path integral formula for the solution to the corresponding heat equation. This is based on approximating path space by finite dimensional spaces of geodesic polygons. We also show a uniform convergence result for the heat kernels. This yields a simple and natural proof for the Hess-Schrader-Uhlenbrock estimate and a path integral formula for the trace of the heat operator.Comment: 23 page

Differential K-theory. A survey

Generalized differential cohomology theories, in particular differential K-theory (often called "smooth K-theory"), are becoming an important tool in differential geometry and in mathematical physics. In this survey, we describe the developments of the recent decades in this area. In particular, we discuss axiomatic characterizations of differential K-theory (and that these uniquely characterize differential K-theory). We describe several explicit constructions, based on vector bundles, on families of differential operators, or using homotopy theory and classifying spaces. We explain the most important properties, in particular about the multiplicative structure and push-forward maps and will state versions of the Riemann-Roch theorem and of Atiyah-Singer family index theorem for differential K-theory.Comment: 50 pages, report based in particular on work done sponsored the DFG SSP "Globale Differentialgeometrie". v2: final version (only typos corrected), to appear in C. B\"ar et al. (eds.), Global Differential Geometry, Springer Proceedings in Mathematics 17, Springer-Verlag Berlin Heidelberg 201

A remark on positively curved 4-manifolds

AbstractLet M be an oriented connected compact Riemannian 4-manifold. We show that if the sectional curvature satisfies K⩾1 and the covariant differential of the curvature tensor satisfies ‖▿R‖L∲⩽2⧸π, then the intersection form of M is definite

Generalized Cylinders in Semi-Riemannian and Spin Geometry

We use a construction which we call generalized cylinders to give a new proof of the fundamental theorem of hypersurface theory. It has the advantage of being very simple and the result directly extends to semi-Riemannian manifolds and to embeddings into spaces of constant curvature. We also give a new way to identify spinors for different metrics and to derive the variation formula for the Dirac operator. Moreover, we show that generalized Killing spinors for Codazzi tensors are restrictions of parallel spinors. Finally, we study the space of Lorentzian metrics and give a criterion when two Lorentzian metrics on a manifold can be joined in a natural manner by a 1-parameter family of such metrics.Comment: 29 pages, 2 figure

Zum Vorhandensein des Zahnmedizinischen Bonusheftes bei Pflegebedürftigen

Ältere und auch pflegebedürftige Menschen werden den Alltag in den Zahnarztpraxen in den kommenden Jahren immer stärker prägen. Mit zunehmender Gebrechlichkeit der Betagten und Hochbetagten sollte sich die zahnmedizinische Behandlung in eine zahnmedizinische Betreuung wandeln. Auch bei einer, im Vergleich zu früher, später eintretenden Pflegebedürftigkeit sind in der Zukunft bei vielen Pflegebedürftigen die zweiten und dritten Zähne zu versorgen. Leider ist festzustellen, dass die zahnmedizinische Versorgung in Pflegeeinrichtungen noch nicht ausreichend standardisiert ist. Von den Pflegebedürftigen in Sachsen, Berlin und Nordrhein Westfalen besaßen nur 18,6 % stationär Pflegebedürftige und 41,2 % ambulant Pflegebedürftige ein zahnmedizinisches Bonusheft. Die Verteilung war regional sehr unterschiedlich. Anteilig besaßen die Pflegebedürftigen in Sachsen am häufigsten ein Bonusheft. Das Vorhandensein des Bonusheftes war weder von Alter oder Geschlecht abhängig. Der Verlust einer besseren Bonusregelung bei Zahnersatzleistungen wird dabei von den Pflegedienstleitungen und dem medizinischen Dienst völlig außer Acht gelassen, teilweise sind die Bonusregelungen auch nicht ausreichend bekannt. Die Organisation eines kontinuierlichen Konsildienstes könnte mit der Anforderung an das Führen eines zahnmedizinischen Bonusheftes verbessert werden. Hierzu müssen aber die Informationen zum Bonusheft gezielt an die Heimleitungen, die Heimaufsichten, die Heimräte, die Angehörigen und an den medizinischen Dienst herangetragen werden

Multiscale Representations for Manifold-Valued Data

We describe multiscale representations for data observed on equispaced grids and taking values in manifolds such as the sphere $S^2$, the special orthogonal group $SO(3)$, the positive definite matrices $SPD(n)$, and the Grassmann manifolds $G(n,k)$. The representations are based on the deployment of Deslauriers--Dubuc and average-interpolating pyramids "in the tangent plane" of such manifolds, using the $Exp$ and $Log$ maps of those manifolds. The representations provide "wavelet coefficients" which can be thresholded, quantized, and scaled in much the same way as traditional wavelet coefficients. Tasks such as compression, noise removal, contrast enhancement, and stochastic simulation are facilitated by this representation. The approach applies to general manifolds but is particularly suited to the manifolds we consider, i.e., Riemannian symmetric spaces, such as $S^{n-1}$, $SO(n)$, $G(n,k)$, where the $Exp$ and $Log$ maps are effectively computable. Applications to manifold-valued data sources of a geometric nature (motion, orientation, diffusion) seem particularly immediate. A software toolbox, SymmLab, can reproduce the results discussed in this paper

Magnetic resonance imaging goes postmortem: noninvasive detection and assessment of myocardial infarction by postmortem MRI

Objective: To investigate the performance of postmortem magnetic resonance imaging (pmMRI) in identification and characterization of lethal myocardial infarction in a non-invasive manner on human corpses. Materials and Methods: Before forensic autopsy, 20 human forensic corpses were examined on a 1.5-T system for the presence of myocardial infarction. Short axis, transversal and longitudinal long axis images (T1-weighted; T2-weighted; PD-weighted) were acquired in situ. In subsequent autopsy, the section technique was adapted to short axis images. Histological investigations were conducted to confirm autopsy and/or radiological diagnoses. Results: Nineteen myocardial lesions were detected and age staged with pmMRI, of which 13 were histologically confirmed (chronic, subacute and acute). Six lesions interpreted as peracute by pmMRI showed no macroscopic or histological finding. Five of the six peracute lesions correlated well to coronary pathology, and one case displayed a severe hypertrophic alteration. Conclusion: pmMRI reliably demonstrates chronic, subacute and acute myocardial infarction in situ. In peracute cases pmMRI may display ischemic lesions undetectable at autopsy and routine histology. pmMRI has the potential to substantiate autopsy and to counteract the loss of reliable information on causes of death due to the recent disappearance of the clinical autops

Spectral Bounds for Dirac Operators on Open Manifolds

We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich's estimate for manifolds with positive scalar curvature as well as the author's estimate on surfaces.Comment: pdflatex, 14 pages, 3 figure