Ebene Äquiforme Zwangläufe im Grossen II

Im ersten Teil dieser Arbeit¹ wurden die orientierten Flächeninhalte der Punktbahnen ebener äquiformer Zwangläufe studiert und einige Anwendungen der erzielten Resultate angeführt. Der vorliegende zweite Teil beschäftigt sich - unter Beibehaltung der Bezeichnungen, der Numerierung der Abschnitte, Sätze usw. - einerseits mit Geradenhüllbahnen, andererseits mit globalen Eigenschaften spezieller äquiformer Zwangläufe, und zwar mit der äquiformen Bewegung der Krümmungsstrecken einer ebenen Kurve, mit äquiformen Konchoiden-bewegungen und mit der äquiformen Bewegung der Durchmesser eines beschränkten konvexen Bereichs. Beim Studium der durch formale Integration gewonnenen, vorzeichenbehafteten “Längen” der Geradenhüllbahnen wurden zur Interpretation des Vorzeichens orientierte Geraden des Gangsystems betrachtet. Dieser anscheinend notwendige Übergang zu Speeren wurde bislang bei der Formulierung von Holditch-Sätzen über Längen von Geradenhüllbahnen (vgl.[11],[39]) nicht beachtet. Wie schon im ersten Teil gestatten die Ergebnisse eine Anwendung in der globalen euklidischen n-Lagentheorie sowie auf nichtgeschlossene euklidische Zwangläufe.     

On Reciprocal Equivalence of Stäckel Systems

In this paper we investigate Stäckel transforms between different classes of parameter‐dependent Stäckel separable systems of the same dimension. We show that the set of all Stäckel systems of the same dimension splits to equivalence classes so that all members within the same class can be connected by a single Stäckel transform. We also give an explicit formula relating solutions of two Stäckel‐related systems. These results show in particular that any two geodesic Stäckel systems are Stäckel equivalent in the sense that it is possible to transform one into another by a single Stäckel transform. We also simplify proofs of some known statements about multiparameter Stäckel transform.     

Einstein almost cokähler manifolds

We study an odd‐dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non‐compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with nonnegative *‐scalar curvature are cokähler (indeed, transversely Calabi–Yau); more generally, we give a lower and upper bound for the *‐scalar curvature in the case that the structure is not cokähler. We prove similar bounds for almost Kähler Einstein manifolds that are not Kähler.     

On the structure of nearly pseudo-Kähler manifolds

Firstly we give a condition to split off the Kähler factor from a nearly pseudo-Kähler manifold and apply this to get a structure result in dimension 8. Secondly we extend the construction of nearly Kähler manifolds from twistor spaces to negatively curved quaternionic Kähler manifolds and para-quaternionic Kähler manifolds. The class of nearly pseudo-Kähler manifolds obtained from this construction is characterized by a holonomic condition. The combination of these results enables us to give a classification result in (real) dimension 10. Moreover, we show that a strict nearly pseudo-Kähler six-manifold is Einstein.     

Locally conformally flat Kähler and para-Kähler manifolds

We complete the classification of locally conformally flat Kähler and para-Kähler manifolds, describing all possible non-flat curvature models for Kähler and para-Kähler surfaces.

     

Left invariant special Kähler structures

We construct left invariant special Kähler structures on the cotangent bundle of a flat pseudo-Riemannian Lie group. We introduce the twisted cartesian product of two special Kähler Lie algebras according to two linear representations by infinitesimal Kähler transformations. We also exhibit a double extension process of a special Kähler Lie algebra which allows us to get all simply connected special Kähler Lie groups with bi-invariant symplectic connections. All Lie groups constructed by performing this double extension process can be identified with a subgroup of symplectic (or Kähler) affine transformations of its Lie algebra containing a nontrivial 1-parameter subgroup formed by central translations. We show a characterization of left invariant flat special Kähler structures using étale Kähler affine representations, exhibit some immediate consequences of the constructions mentioned above, and give several non-trivial examples.     

On Kähler manifolds with harmonic Bochner curvature tensor

We prove that every irreducible Kähler manifold with harmonic Bochner curvature tensor and constant scalar curvature is Kähler–Einstein and that every irreducible compact Kähler manifold with harmonic Bochner curvature tensor and negative semi-definite Ricci tensor is Kähler–Einstein.     

Quaternionic Kähler manifolds with Hermitian and Norden metrics

Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kähler manifolds are considered. Some necessary and sufficient conditions for the investigated manifolds to be isotropic hyper-Kählerian and flat are found. It is proved that the quaternionic Kähler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kähler quaternionic Kähler manifolds of the considered type is determined.     

Kähler manifolds of quasi-constant holomorphic sectional curvatures

The Kähler manifolds of quasi-constant holomorphic sectional curvatures are introduced as Kähler manifolds with complex distribution of codimension two, whose holomorphic sectional curvature only depends on the corresponding point and the geometric angle, associated with the section. A curvature identity characterizing such manifolds is found. The biconformal group of transformations whose elements transform Kähler metrics into Kähler ones is introduced and biconformal tensor invariants are obtained. This makes it possible to classify the manifolds under consideration locally. The class of locally biconformal flat Kähler metrics is shown to be exactly the class of Kähler metrics whose potential function is only a function of the distance from the origin in ? ⁿ . Finally we show that any rotational even dimensional hypersurface carries locally a natural Kähler structure which is of quasi-constant holomorphic sectional curvatures.     

Decomposition and minimality of lagrangian submanifolds in nearly Kähler manifolds

We show that Lagrangian submanifolds in six-dimensional nearly Kähler (non-Kähler) manifolds and in twistor spaces Z ⁴ⁿ⁺² over quaternionic Kähler manifolds Q ⁴ⁿ are minimal. Moreover, we prove that any Lagrangian submanifold L in a nearly Kähler manifold M splits into a product of two Lagrangian submanifolds for which one factor is Lagrangian in the strict nearly Kähler part of M and the other factor is Lagrangian in the Kähler part of M. Using this splitting theorem, we then describe Lagrangian submanifolds in nearly Kähler manifolds of dimensions six, eight, and ten.     

Nearly Kähler Submanifolds of a Space Form

In this article, we study isometric immersions of nearly Kähler manifolds into a space form (especially Euclidean space) and show that every nearly Kähler submanifold of a space form has an umbilic foliation whose leafs are 6-dimensional nearly Kähler manifolds. Moreover, using this foliation we show that there is no non-homogeneous 6-dimensional nearly Kähler submanifold of a space form. We prove some results towards a classification of nearly Kähler hypersurfaces in standard space forms.     

On a Class of Almost-Hermitian Structures on Tangent Bundles

We construct a new almost-Hermitian structure of anti-invariant type on tangent bundles and deduce criteria for this structure to belong to all the Gray--Hervella classes. In particular, we prove that the tangent bundles over Kählerian and semi-Kählerian manifolds carry, respectively, a Kählerian and a semi-Kählerian structure.     

Wilhelm Blaschke (13.09.1885-17.03.1962)

Gerne folge ich dem Wunsche der Redaktion der “Resultate der Mathematik” anläßlich der hundertsten Wiederkehr des Tages der Geburt von Wilhelm Blaschke im September 1985 eine Darstellung des Lebens und Wirkens dieses großen Geometers zu geben. Ich komme dieser Aufforderung um so lieber nach, weil ich Blaschke stets als einen väterlichen Freund geschätzt und verehrt habe, dessen geometrischem Schaffen ich viele Anregungen verdanke. Die Beziehungen Blaschkes zu der Karlsruher Technischen Hochschule (seit 1967 Universität Karlsruhe) waren demgemäß auch immer sehr eng; Blaschke war häufiger Vortragsgast in unserem mathematischen Kolloquium und seit 1960 Karlsruher Ehrendoktor.     

Numerische Simulation der Freispiegelströmung in einem Wasserrad

Es wird ein Berechnungsmodell für 2-Fluid-Strömungen mit drehendem Starrkörper vorgestellt, mit dem eine numerische Analyse der Strömungsvorgänge innerhalb von Schaufelwasserrädern möglich ist. Mit dem vorgestellten Berechnungsmodell erfolgt die erstmalige numerische Untersuchung der bei Schaufelwasserrädern auftretenden Strömungsvorgänge. Dabei werden die wesentlichen Punkte im Rahmen der numerischen Umsetzung eingehend diskutiert und die gewonnenen Ergebnisse anhand experimenteller Daten auf ihre physikalische Plausibilität hin überprüft. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Convergence of Kähler-Ricci flow

In this paper, we prove a theorem on convergence of Kähler-Ricci flow on a compact Kähler manifold which admits a Kähler-Ricci soliton.

     

Almost contact Kähler manifolds of constant holomorphic sectional curvature

We introduce the notion of an almost contact Kähler structure. We also define the holomorphic sectional curvature of the distribution of an almost contact Kähler structure with respect to an interior metric connection and establish relations between the φ-sectional curvature of an almost contact Kähler manifold and the holomorphic sectional curvature of the distribution of an almost contact Kähler structure.     

HIGH ORDER KÄHLER MODULES OF NONCOMMUTATIVE RING EXTENSIONS

We construct the high order Kähler modules of noncommutative ring extensions B/A and show their fundamental properties. Our Kähler modules represent not only high order left derivations for one-sided modules but also high order central derivations for bimodules, which are usual derivations. This new viewpoint enables us to prove new results which were not known even though B is an algebra over a commutative ring A. Our results are the decomposition of Kähler modules by an idempotent element, exact sequences of Kähler modules, the Kähler modules of factor rings, and the relation to separable extensions. In particular, our exact sequences of high order Kähler modules were not known even though B is commutative.     

Regularity of the Kähler–Ricci flow

In this short note, we announce a regularity theorem for the Kähler–Ricci flow on a compact Fano manifold (Kähler manifold with positive first Chern class) and its application to the limiting behavior of the Kähler–Ricci flow on Fano 3-manifolds. Moreover, we also present a partial $C^0$ estimate of the Kähler–Ricci flow under the regularity assumption, which extends previous works on Kähler–Einstein metrics and shrinking Kähler–Ricci solitons. The detailed proof will appear elsewhere.