Spectral flow and iteration of closed semi-Riemannian geodesics

We introduce the notion of spectral flow along a periodic semi-Riemannian geodesic, as a suitable substitute of the Morse index in the Riemannian case. We study the growth of the spectral flow along a closed geodesic under iteration, determining its asymptotic behavior.

GENERICITY OF NONDEGENERATE GEODESICS WITH GENERAL BOUNDARY CONDITIONS

Let M be a possibly noncompact manifold. We prove, generically in the C(k)-topology (2 <= k <= infinity), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of L. Biliotti, M. A. Javaloyes and P. Piccione [6] for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P subset of M x M that satisfies an admissibility condition. Such condition holds, for example, when P is transversal to the diagonal Delta subset of M x M. Further aspects of these boundary conditions are discussed and general conditions under which metrics without degenerate geodesics are C(k)-generic are given.

On the semi-Riemannian bumpy metric theorem

We prove the semi-Riemannian bumpy metric theorem using equivariant variational genericity. The theorem states that, on a given compact manifold M, the set of semi-Riemannian metrics that admit only nondegenerate closed geodesics is generic relatively to the C(k)-topology, k=2, ..., infinity, in the set of metrics of a given index on M. A higher-order genericity Riemannian result of Klingenberg and Takens is extended to semi-Riemannian geometry.

Maslov index in semi-Riemannian submersions

We study focal points and Maslov index of a horizontal geodesic gamma : I -> M in the total space of a semi-Riemannian submersion pi : M -> B by determining an explicit relation with the corresponding objects along the projected geodesic pi omicron gamma : I -> B in the base space. We use this result to calculate the focal Maslov index of a (spacelike) geodesic in a stationary spacetime which is orthogonal to a timelike Killing vector field.

COMPARISON RESULTS FOR CONJUGATE AND FOCAL POINTS IN SEMI-RIEMANNIAN GEOMETRY VIA MASLOV INDEX

We prove an estimate on the difference of Maslov indices relative to the choice of two distinct reference Lagrangians of a continuous path in the Lagrangian Grassmannian of a symplectic space. We discuss some applications to the study of conjugate and focal points along a geodesic in a semi-Riemannian manifold.

NEW CHARACTERIZATIONS OF COMPLETE SPACELIKE SUBMANIFOLDS IN SEMI-RIEMANNIAN SPACE FORMS

In this paper we study n-dimensional complete spacelike submanifolds with constant normalized scalar curvature immersed in semi-Riemannian space forms. By extending Cheng-Yau`s technique to these ambients, we obtain results to such submanifolds satisfying certain conditions on both the squared norm of the second fundamental form and the mean curvature. We also characterize compact non-negatively curved submanifolds in De Sitter space of index p.

Associated Family of G-Structure Preserving Minimal Immersions in Semi-Riemannian Manifolds

We prove the existence of an associated family of G-structure preserving minimal immersions into semi-Riemannian manifolds endowed with a compatible infinitesimally homogeneous G-structure. We will study in more details minimal embeddings into product of space forms.

Some Para-Hermitian Related Complex Structures and Non-existence of Semi-Riemannian Metric on Some Spheres

It is shown that the spheres S^(2n) (resp: S^k with k ≡ 1 mod 4) can be given neither an indefinite metric of any signature (resp: of signature (r, k − r) with 2 ≤ r ≤ k − 2) nor an almost paracomplex structure. Further for every given Riemannian metric on an almost para-Hermitian manifold with the associated 2-form φ one can construct an almost Hermitian structure (under certain conditions, two different almost Hermitian structures) whose associated 2-form(s) is φ.

On a formula for the spectral flow and its applications

We consider a continuous path of bounded symmetric Fredholm bilinear forms with arbitrary endpoints on a real Hilbert space, and we prove a formula that gives the spectral flow of the path in terms of the spectral flow of the restriction to a finite codimensional closed subspace. We also discuss the case of restrictions to a continuous path of finite codimensional closed subspaces. As an application of the formula, we introduce the notion of spectral flow for a periodic semi-Riemannian geodesic, and we compute its value in terms of the Maslov index. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Genericity of Nondegenerate Critical Points and Morse Geodesic Functionals

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Using classical techniques, we prove an abstract genericity result that employs the infinite dimensional Sard-Smale theorem, along the lines of an analogous result of B. White [29]. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.

Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Given a Lorentzian manifold (M,g), a geodesic gamma in M and a timelike Jacobi field Y along gamma, we introduce a special class of instants along gamma that we call Y-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Y-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Y is obtained as the restriction of a globally defined timelike Killing vector field.

Isometric immersions into a homogeneous Lorentzian Heisenberg group and rigidity

In this paper we prove an existence result for local and global isometric immersions of semi-Riemannian surfaces into the three dimensional Heisenberg group endowed with a homogeneous left-invariant Lorentzian metric. As a corollary, we prove a rigidity result for such immersions.

An existence theorem for G-structure preserving affine immersions

We prove an existence result for local and global G-structure preserving affine immersions between affine manifolds. Several examples are discussed in the context of Riemannian and semi-Riemannian geometry, including the case of isometric immersions into Lie groups endowed with a left-invariant metric, and the case of isometric immersions into products of space forms.

Tenseness of Riemannian flows

We show that any transversally complete Riemannian foliation &em&F&/em& of dimension one on any possibly non-compact manifold M is tense; namely, (M,&em&F&/em&) admits a Riemannian metric such that the mean curvature form of &em&F&/em& is basic. This is a partial generalization of a result of Domínguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Domínguez. As an application, we generalize some well known results including Masa's characterization of tautness.

Impact of sward type, cutting date and conditioning temperature on mass and energy flows in the integrated generation of solid fuel and biogas from semi-natural grassland

Energy production from biomass and the conservation of ecologically valuable grassland habitats are two important issues of agriculture today. The combination of a bioenergy production, which minimises environmental impacts and competition with food production for land with a conversion of semi-natural grasslands through new utilization alternatives for the biomass, led to the development of the IFBB process. Its basic principle is the separation of biomass into a liquid fraction (press fluid, PF) for the production of electric and thermal energy after anaerobic digestion to biogas and a solid fraction (press cake, PC) for the production of thermal energy through combustion. This study was undertaken to explore mass and energy flows as well as quality aspects of energy carriers within the IFBB process and determine their dependency on biomass-related and technical parameters. Two experiments were conducted, in which biomass from semi-natural grassland was conserved as silage and subjected to a hydrothermal conditioning and a subsequent mechanical dehydration with a screw press. Methane yield of the PF and the untreated silage was determined in anaerobic digestion experiments in batch fermenters at 37°C with a fermentation time of 13-15 and 27-35 days for the PF and the silage, respectively. Concentrations of dry matter (DM), ash, crude protein (CP), crude fibre (CF), ether extract (EE), neutral detergent fibre (NDF), acid detergent fibre (ADF), acid detergent ligning (ADL) and elements (K, Mg, Ca, Cl, N, S, P, C, H, N) were determined in the untreated biomass and the PC. Higher heating value (HHV) and ash softening temperature (AST) were calculated based on elemental concentration. Chemical composition of the PF and mass flows of all plant compounds into the PF were calculated. In the first experiment, biomass from five different semi-natural grassland swards (Arrhenaterion I and II, Caricion fuscae, Filipendulion ulmariae, Polygono-Trisetion) was harvested at one late sampling (19 July or 31 August) and ensiled. Each silage was subjected to three different temperature treatments (5°C, 60°C, 80°C) during hydrothermal conditioning. Based on observed methane yields and HHV as energy output parameters as well as literature-based and observed energy input parameters, energy and green house gas (GHG) balances were calculated for IFBB and two reference conversion processes, whole-crop digestion of untreated silage (WCD) and combustion of hay (CH). In the second experiment, biomass from one single semi-natural grassland sward (Arrhenaterion) was harvested at eight consecutive dates (27/04, 02/05, 09/05, 16/05, 24/05, 31/05, 11/06, 21/06) and ensiled. Each silage was subjected to six different treatments (no hydrothermal conditioning and hydrothermal conditioning at 10°C, 30°C, 50°C, 70°C, 90°C). Energy balance was calculated for IFBB and WCD. Multiple regression models were developed to predict mass flows, concentrations of elements in the PC, concentration of organic compounds in the PF and energy conversion efficiency of the IFBB process from temperature of hydrothermal conditioning as well as NDF and DM concentration in the silage. Results showed a relative reduction of ash and all elements detrimental for combustion in the PC compared to the untreated biomass of 20-90%. Reduction was highest for K and Cl and lowest for N. HHV of PC and untreated biomass were in a comparable range (17.8-19.5 MJ kg-1 DM), but AST of PC was higher (1156-1254°C). Methane yields of PF were higher compared to those of WCD when the biomass was harvested late (end of May and later) and in a comparable range when the biomass was harvested early and ranged from 332 to 458 LN kg-1 VS. Regarding energy and GHG balances, IFBB, with a net energy yield of 11.9-14.1 MWh ha-1, a conversion efficiency of 0.43-0.51, and GHG mitigation of 3.6-4.4 t CO2eq ha-1, performed better than WCD, but worse than CH. WCD produces thermal and electric energy with low efficiency, CH produces only thermal energy with a low quality solid fuel with high efficiency, IFBB produces thermal and electric energy with a solid fuel of high quality with medium efficiency. Regression models were able to predict target parameters with high accuracy (R2=0.70-0.99). The influence of increasing temperature of hydrothermal conditioning was an increase of mass flows, a decrease of element concentrations in the PC and a differing effect on energy conversion efficiency. The influence of increasing NDF concentration of the silage was a differing effect on mass flows, a decrease of element concentrations in the PC and an increase of energy conversion efficiency. The influence of increasing DM concentration of the silage was a decrease of mass flows, an increase of element concentrations in the PC and an increase of energy conversion efficiency. Based on the models an optimised IFBB process would be obtained with a medium temperature of hydrothermal conditioning (50°C), high NDF concentrations in the silage and medium DM concentrations of the silage.