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 $\mathcal Y$ along $\gamma$, we introduce a special class of instants along $\gamma$ that we call $\mathcal Y$-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the $\mathcal Y$-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 $\mathcal Y$ is obtained as the restriction of a globally defined timelike Killing vector field.Comment: 26 pages, Proposition 3.10, that has become Proposition 3.11 in the current version, has changed introducing a new class of points that we call pseudo-focal points. The new Section 5 is devoted to describe some properties to these point

A petrofabric study of the iron ore of Benson Mines, Star Lake, New York

The petrofabrics of the host gneiss silicates and Ovate ore grains were studied in thin sections of oriented specimens collected from widely scattered positions in the iron ore deposit at Benson Mines, New York. Ten oriented specimens selected for petrofabrics study were comprised primarily of feldspar, quartz, garnet and ores, with lesser quantities of biotite and sillimanite. Petrofabric diagrams were prepared for the orientation fabrics of magnetite and biotite in eight of the ten specimens, but sillimanite and quartz were sufficiently abundant for fabric determination in only three specimens. Duplicate thin section and perpendicular sections were utilized to check the resultant orientation patterns. A total of 28 petrofabric diagrams were prepared. The petrofabric diagrams for the basal cleavage of biotite grains reveals foliation which is much more distinct than that observed rnegascopically. The c-axes of sillimanite are distinctly aligned in a direction of lineation which lies within the biotite foliation plane. The petrofabric diagrams for the long axes of ovate ore grains exhibit more scatter than those of biotite and sillimanite, but those axes exhibit a tendency to lie within the biotite foliation plane and to be relatively concentrated in the direction of sillimanite lineation. This tendency is strongest in the more strongly foliated and lineated specimens respectively. The correspondence between host gneiss silicate petrofabrics and orientation of ovate ore grains suggests that the iron oxide grains have developed preferred orientation during the metamorphism which developed the foliation in the host rock gneiss and that the iron was introduced or simply present in the rock prior to the last stages of metamorphism --Abstract, pages ii-iii

Harmonic Coordinates for the Nonlinear Finsler Laplacian and Some Regularity Results for Berwald Metrics

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers--Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we obtain some partial results in this direction when the Finsler metric is Berwald.Comment: AMS-LaTeX, 11 page

A Morse Theory for Massive Particles and Photons in General Relativity

In this paper we develop a Morse Theory for timelike geodesics parameterized by a constant multiple of proper time. The results are obtained using an extension to the timelike case of the relativistic Fermat Principle, and techniques from Global Analysis on infinite dimensional manifolds. In the second part of the paper we discuss a limit process that allows to obtain also a Morse theory for light rays

Connecting and closed geodesics of a Kropina metric

We prove some results about existence of connecting and closed geodesics in a manifold endowed with a Kropina metric. These have applications to both null geodesics of spacetimes endowed with a null Killing vector field and Zermelo's navigation problem with critical wind.Comment: 15 pages, AMSLaTe