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We prove Gaus-Bonnet-type and Chern-Lashof-type formulas for immersions in hyperbolic space. Moreover we investigate the notion of tightness with respect to horospheres introduced by T.E. Cecil and P.J. Ryan. We introduce the notions of top-set and drop-set, and we prove fundamental properties of horo-tightness in hyperbolic spaces.

In this article we introduce and investigate linear combinations of hypersurfaces in hyperbolic space. For this purpose we use some linear structure in the space of horospheres.

Given two surfaces in three dimensional euclidean space, one fixed and the other moved by rigid motions, we consider the total absolute curvature of the intersection curves. In this paper we investigate the integral of these total absolute curvatures over all motions. Under some geometric conditions we obtain kinematic formulas, and with weaker conditions we get upper and lower bounds. Finally, as applications, we obtain upper bounds for the average number of connected components of the intersections, and we give Hadwiger conditions for a convex domain to be able to contain another one.

We consider the measure of points, the measure of lines and the measure of planes intersecting a given convex body K in a space form. We obtain some integral formulas involving the width of K and the curvature of its boundary ∂K. Also we study the special case of constant width. Moreover we obtain a generalisation of the Heintze–Karcher inequality to space forms.

We extend classical Cauchy formulas and Crofton formulas to the Lorentz-Minkowski space, and to constant curvature Lorentz spaces.

where the curve is given by a smooth immersion f of a circle into E and w I (p) denotes the winding number of f w.r.t, p ~ E. Equality holds if and only if the curve is a circle traversed in the same direction a number of times. (In particular Banchoff and Pohl derived an inequality of this type which applies to compact immersed submanifolds in euclidean spaces of arbitrary dimensions.) (Cf. also [3] for a polygonal version of (2).) In the spherical case (1) can be written more symmetrically, namely

Given two curves in the real affine plane, one is fixed and the other undergoes volume-preserving affinities. Through transversal affinities we define a contact measure on the subset consisting of those affinities, which cause third-order contact between the fixed and the transformed curve. A kinematic formula expresses this contact measure in terms of affine lengths and affine curvatures of the given curves. In a similar way, parallel supporting planes of closed convex surfaces in affine space are treated.

The total absolute (Lipschitz-Killing-)curvature of an immersion of a manifold in Euclidean space is equal to the mean value of the number of critical points of certain level functions which are induced by bundles of parallel hyperplanes (see Chern-Lashof and Kuiper). We generalize this for immersions in spaces of constant curvature + 1. A level function is induced by an oriented hyperplane bundle either through a common plane of codimension two or orthogonal to a geodesic. In the hyperbolic case we have to restrict the level function to the interior of the normal tube of constant radius around that plane or geodesic.

1,3-Diphenylbenzo[c]furan (1) reagiert thermisch mit 3,4-Dimethylencyclobuten (2) und 1,3,5-Cycloheptatrien (5) zu exo- und endo-Diels-Alder-Addukten. Die Reaktion von 1 mit Kaliumcycloheptatrienid fuhrt uber das Anionzwischenprodukt 12 bei tiefer Temperatur zu Substitutionsaddukten (13, 14), bei hoherer Temperatur dagegen zum [4 + 2]-Ringschlus (15). Die Photoaddition von 1 an 5 fuhrt abweichend von Literaturangaben zu den exo- und endo-[4 + 2]-, endo-[4 + 6]-, exo-[4 + 4]- und cis-1,4(1,3)-Substitutions-Addukten 7, 6, 16, 17 und 14, deren Konstitution 1H-NMR-spektroskopisch sowie z. T. durch chemische Umwandlungen belegt wird und deren Bildung auf der Grundlage des bewahrten Diradikalmechanismus gedeutet werden kann. Die endo-/exo-Isomerisierung von 16 zu 19 gelingt solvolytisch. Durch suprafaciale thermische 1,3-Alkylverschiebungen entstehen aus 17 die Produkte 7 und 16. Weitere exo- und endo-[4 + 2]- sowie [4 + 4]-Additionen von 1 werden mit Maleinsaure-dimethylester (23, 24) und Anthracen (26) erreicht. Cyclovinylogous Additions of 1,3-Diphenylbenzo[c]furan to 1,3,5-Cycloheptatriene 1,3-Diphenylbenzo[c]furan (1) reacts thermally with 3,4-dimethylenecyclobutene (2) and 1,3,5-cycloheptatriene (5) to give exo- and endo-Diels-Alder adducts. The reaction of 1 with potassium cycloheptatrienide leads to substitutive adducts (13, 14) at low temperature, but to [4 + 2]-ring closure (15) at higher temperatures, via the anionic intermediate 12. The photoaddition of 1 to 5 gives the exo- and endo-[4 + 2]-, endo-[4 + 6]-, exo-[4 + 4]- und cis-1,4(1,3)-substitutive adducts 7, 6, 16, 17 und 14, in divergence to recent literature data. The structures of the products are elucidated by 1H NMR spectra and in part by chemical transformations, their formation can be interpreted in terms of the approved diradical mechanism. The endo-/exo-isomerization of 16 into 19 proceeds solvolytically. The products 7 and 16 are formed out of 17 by suprafacial thermal 1,3-alkyl shifts. Further exo- und endo-[4 + 2]- as well as [4 + 4]-additions of 1 are achieved with dimethyl maleate (23, 24) and anthracene (26), resp.

The aim of this article is to compute the total (absolute) curvature, i.e. the mean value of the (absolute) Lipschitz-Killing-curvature, of an immersion f: M→Sn of a compact manifold into the unit sphere in a differential topological manner. Through a generalization of KUIPERs treatment of immersions in Euclidean spaces it can be computed as the mean value of the number of critical points—weighted by (−1)k (k=Index) resp.not weighted—of certain functions. These functions are the pullback via f of “level-functions”, which are defined almost everywhere on Sn. Such a “level-function” is constructed by taking any oriented great circle as a “leveling-scale” and the orthogonal great (n−1)-spheres as “level-surfaces”.

Elektronisch angeregtes 9-Anthracencarbonitril (1*) wird von 2,4-Hexadien (2) entgegen theoretischen Erwartungen zum [4 + 4]-Addukt 3 mit trans-Doppelbindung abgefangen, aus dem oberhalb 0°C durch suprafaciale 1,3-Verschiebung das [4 + 2]-Addukt 4 entsteht. Auch 1,3,5-Cycloheptatrien (6) reagiert mit 1* zu einem thermolabilen [4 + 4]-Addukt 7 (→ 8). Daneben entstehen weitere Addukte (9, 10, 11, 12) uber isomere Diradikale. Anthracen (19*) bildet mit 6 die [4 + 6]-, [4 + 4]-, [4 + 2]-, Substitutions- und Peripher-Addukte 20, 21, 22, 23, 24. Mit 9-Phenylanthracen (25*) und 6 gibt es die entsprechenden, z. T. stellungsisomeren, Addukte (26; 29; 27, 28; 30, 31; 32). Die Reaktionen werden an Hand von Diradikalmechanismen gedeutet. Zahlreiche bereits theoretisch klassifizierte Literaturangaben werden erganzt oder berichtigt, die Grunde fur die fehlerhaften theoretischen Vorhersagen diskutiert. Die Produktevielfalt kann durch empirische Extrapolation im Rahmen des experimentell gesicherten Cyclovinylogieprinzips detailliert verstanden werden.

1,4-Addition of Potassium Cycloheptatrienide to (9-Phenyl-)Anthracene In liquid ammonia the cycloheptatriene anion adds to anthracene (3) and 9-phenylanthracene (9) to give the anions 4 and 10, resp., which are quenched at −35°C to give the substitutive 1,4-adducts 5 and 11 + 12, resp. Upon heating up to 20°C the intermediate anion 4 cyclizes to the Diels-Alder product 6, which is quenched to give the asymmetric and symmetric [4 + 2]-adducts 7 and 8. The structures of the products have been established primarily by high-field 1H-NMR spectroscopy.

In a previous article [16] we have shown how the total absolute (Lipschitz-Killing) curvature of the immersion f:M→Sn of a compact manifold into a sphere can be computed in a differential topological manner as the mean-value of the number of critical points of certain level-functions. (And similar the total curvature.) (comp. [12], prob. 15) Now we consider the gradient vector field of the level-functions and achieve a relation between the total curvature and the Euler characteristic, of the manifold, which can be sharpened in some cases to inequalities. Moreover it leads to the formula of Allendoerfer-Weil for compact n-dim. submanifolds of the sphere Sn.

Diene-Catalyzed Photodimerization via Excimers (Exciplexes) or Diene Addition to Anthracene and 9-Phenylanthracene The [4 + 4]-photodimerization of anthracene (1) is not enhanced by 2,4-hexadiene (3) but impeded due to formation of [4 + 2]- (4) and [4 + 4]-adducts (5, 6). 6 and 1 combine to give the bisadduct 7 at room temperature. The selective photolysis of 9-phenylanthracene (8) leads to the anti-[4 + 4]-dimer (9). This can be compared with the [4 + 4]-photodimerization of 9-methylanthracene which affords the anti- and the thermolabile syn-product. 1,3-Dienes such as 1,3-pentadiene (10) or 1,3-cyclohexadiene (13) are not necessary for the formation of 9 but undergo cycloadditions with 8 to yield [4 + 2]- (11, 12, 14), [4 + 4]- (15), and [2 + 2]-adducts (16, 17). The structures of the products are established primarily by high field 1H-NMR spectroscopy.

The total absolute curvature of a closed curve in a Euclidean space is always greater or equal to 2 (Fenchel's inequality,1929, [3], [1], [13]); especially for a knotted curve it is always greater than 4 (Fary-Milnor's inequality, 1949, [2], [7], [5], [4]).