Riemannian submersions and the regular interval theorem of Morse theory

A generalized version of the regular interval theorem of Morse theory is proven using techniques from the theory of Riemannian submersions and conformal deformations. This approach provides an interesting link between Riemannian submersions (for real valued functions) and Morse theory.Let % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]: (M,) R be a smooth real valued function on a non-compact complete connected Riemannian manifold (M,g) such that df is bounded in norm away from zero. By pointwise conformally deforming g to pg, p = d% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]², we show that (M,pg) is a complete Riemannian manifold, and that % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]: (M,pg) R is a surjective Riemannian submersion and a globally trivial fiber bundle over R. In particular, all of the level hypersurfaces of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] are diffeomorphic, and M is globally diffeomorphic to the product bundle R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] ^–1(0) by a diffeomorphism F ⁰: R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0) M that straightens out the level hypersurfaces of % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\].Moreover, we show that (F ⁰)^*(pg) is a parameterized Riemannian product manifold on R×% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0), i.e., a product manifold with a metric that varies on the fibers {t} × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0). Also, F ⁰: (R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0),(F ⁰)^*(pg)) (M,g) is a conformal diffeomorphism between the Reimannian manifolds (R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0), (F ⁰)^*(pg)) and (M,g),so that (M,g) is conformally equivalent to a parameterized Riemannian product manifold. The conformal diffeomorphism F ⁰ is an isometry between the Riemannian product manifold (R × % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0), 1 + g ₀) (where g ₀) is the metric induced by g on % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\]^–1(0) and (M,g) if and only if d% MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] = 1 and Hess % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWexLMBb50ujb% qegi0BVTgib5gDPfxDHbacfaGae8NKbmiaaa!3E95!\[f\] = 0.     

Quantum conformal superspace

For a compact connected orientablen-manifoldM, n 3, we study the structure ofclassical superspace ,quantum superspace ,classical conformal superspace , andquantum conformal superspace . The study of the structure of these spaces is motivated by questions involving reduction of the usual canonical Hamiltonian formulation of general relativity to a non-degenerate Hamiltonian formulation, and to questions involving the quantization of the gravitational field. We show that if the degree of symmetry ofM is zero, thenS,S ₀,C, andC ₀ areilh orbifolds. The case of most importance for general relativity is dimensionn=3. In this case, assuming that the extended Poincaré conjecture is true, we show that quantum superspaceS ₀ and quantum conformal superspaceC ₀ are in factilh-manifolds. If, moreover,M is a Haken manifold, then quantum superspace and quantum conformal superspace arecontractible ilh-manifolds. In this case, there are no Gribov ambiguities for the configuration spacesS ₀ andC ₀. Our results are applicable to questions involving the problem of thereduction of Einstein's vacuum equations and to problems involving quantization of the gravitational field. For the problem of reduction, one searches for a way to reduce the canonical Hamiltonian formulation together with its constraint equations to an unconstrained Hamiltonian system on a reduced phase space. For the problem of quantum gravity, the spaceC ₀ will play a natural role in any quantization procedure based on the use of conformal methods and the reduced Hamiltonian formulation.     

Crystal and Molecular Structure of an Iodo-derivative of the Cyclic Undecapeptide Cyclosporin A

The crystal structure of an iodo-derivative of cyclosporin A has been determined in order to elucidate the constitution of this cyclic undecapeptide. Crystals of iodocyclosporin A are monoclinic, a = 10.475(5), b = 19.60(1), c = 21.04(1) Å, β 99.35(2)°, space group P2₁ (C, No. 4). The structure was solved by the heavy atom method and refined by block-diagonal least-squares analysis to a final R-factor of 0.135 with hydrogen atoms in calculated positions. The cyclic peptide has a conformation which is partly β-pleated sheet and partly open loop. The structure analysis demonstrates for the first time the reality of a new type of dipeptide hydrogen-bonding, predicted by Pullman from MO calculations and leading to a conformation described by him as C.     

Vinylogous urethanes in alkaloid synthesis: formal syntheses of Elaeocarpus alkaloids

The substituted dehydroindolizidine $\text{8}$ has been prepared from the exocyclic vinylogous urethane $\text{7}$ via an acylative ring closure. The specific formation of the enolate required for further elaboration to Elaeocarpus alkaloid precursors has been achieved by making use of the vinylogous amide $\text{12}$.     

Charge separation in heterostructures of InP nanocrystals with metal particles

The optical and electron paramagnetic resonance (EPR) properties of InP nanocrystals, in which metallic gold or indium is present as an incorporated part of the nanocrystals, have been studied. A study of Au/InP quantum rods supports different carrier localization regimes compared to metal-free quantum rods, including the charge-separated state for which the electron and hole are located in different parts of the heterostructure. They also show that elongated semiconductors that grow on metallic catalysts have electronic properties that are different from those of pure semiconductor nanocrystals of the same shape. We have also developed a simple method for growing melted indium particles on the surface of colloidal spherical InP nanocrystals, and in these In/InP nanocrystals the emission is completely quenched while the absorption spectrum moves to red due to the strong mixing of the semiconductor and metal electronic states.     

Vibrationally mediated photodissociation of ethylene cation by reflectron multimass velocity map imaging

A new imaging technique, reflectron multimass velocity map ion imaging, is used to study the vibrationally mediated photodissociation dynamics in the ethylene cation. The cation ground electronic state is prepared in specific vibrational levels by two-photon resonant, three-photon ionization via vibronic bands of (pi, nf) Rydberg states in the vicinity of the ionization potential of ethylene, then photodissociated through the (B 2A(g)) excited state. We simultaneously record spatially resolved images of parent C2H4+ ions as well as photofragment C2H3+ and C2H2+ ions originating in dissociation from the vibronic excitations in two distinct bands, 7f 4(0)2 and 8f 0(0)0, at roughly the same total energy. By analyzing the images, we directly obtain the total translation energy distributions for the two dissociation channels and the branching between them. The results show that there exist differences for competitive dissociation pathways between H and H2 elimination from C2H4+ depending on the vibronic preparation used, i.e., on the vibrational excitation in the ground state of the cation prior to photodissociation. Our findings are discussed in terms of the possible influence of the torsional excitation on competition between direct dissociation, isomerization, and radiationless transitions through conical intersections among the numerous electronic states that participate in the dissociation.     

Novel fragmentation reaction of 2-alkyl- and 2,4-dialkyl-3-iodo-1-oxocyclohexan-2,4-carbolactones

2-Alkyl- and 2,4-dialkyl-3-iodo-1-oxocyclohexan-2,4-carbolactones undergo lithium hydroxide- and lithium alkoxide-induced fragmentation reactions to provide butenolides, gamma-hydroxycyclohexenones, and/or gamma-butyrolactones. In general, product distribution is governed by two factors: (1) the nature of nucleophiles and (2) the steric bulkiness of the substituents at C-2 and C-4 of the cyclohexanones. Lithium hydroxide-induced fragmentation provides butenolides and gamma-hydroxycyclohexenones. In contrast, lithium alkoxide-promoted fragmentation results in predominantly 5-substituted gamma-butyrolactones along with a small amount of butenolides in limited cases. Fragmentation products induced by lithium hydroxide are largely influenced by the steric bulkiness of the substituents at C-2 and C-4 of the cyclohexanone ring. The bulky substituents render the exclusive formation of butenolides.     

Adsorption of a phospholipid-hydroperoxide glutathione peroxidase into phospholipid monolayers at the air-water interface

The interfacial behavior differences of two glutathione peroxidase isoforms have been investigated. The first isoform is the phospholipid-hydroperoxide glutathione peroxidase (EC 1.11.1.12) (GPx-4) isolated from rat testes and the second one is the cytosolic glutathione peroxidase (EC 1.11.1.9) (GPx-1) from bovine erythrocytes. Injected in the subphase buffer of a Langmuir trough, GPx-4 was able to adsorb quickly at the air-water interface whereas the GPx-1 was not. Then, the protein interaction with phospholipid monolayers was explored. Indeed, a monolayer of phospholipids containing a different number of polyunsaturated fatty acyl chains was prepared at the air-water interface. Under each kind of monolayer, the protein solution was injected and its adsorption was visualized by the measurement of successive pressure-area isotherms. We have, then, determined the molecular area increase due to the protein adsorption. It was found that the GPx-4 is adsorbed in each kind of monolayer tested whereas no molecular area increase was detected with the GPx-1. This indicates that the GPx-4 has a higher affinity for the interface, recovered or not by lipids, than the GPx-1. Moreover, the GPx-4 presents a different affinity for the phospholipid monolayers depending on the number of polyunsaturated fatty acyl chains.