Paolo Piccione

The Morse index theorem in semi-Riemannian geometry

Abstract We prove a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics in semi-Riemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the Maslov index of a semi-Riemannian geodesic, which is a homological invariant and it substitutes the notion of geometric index in Riemannian geometry. Under generic circumstances, the Maslov index of a geodesic is computed as a sort of algebraic count of the conjugate points along the geodesic. For nonpositive definite metrics the index of the index form is always infinite; in this paper we prove that the space of all variations of a given geodesic has a natural splitting into two infinite dimensional subspaces, and the Maslov index is given by the difference of the index and the coindex of the restriction of the index form to these subspaces. In the case of variable endpoints, two suitable correction terms, defined in terms of the endmanifolds, are added to the equality. Using appropriate change of variables, the theory is entirely extended to the more general case of symplectic differential systems , that can be obtained as linearizations of the Hamilton equations. The main results proven in this paper were announced in Piccione and Tausk (C. R. Acad. Sci. Paris 331 (5) (2000) 385).

New Solutions of Einstein Equations in Spherical Symmetry: The Cosmic Censor to the Court

Abstract: A new class of solutions of the Einstein field equations in spherical symmetry is found. The new solutions are mathematically described as the metrics admitting separation of variables in area-radius coordinates. Physically, they describe the gravitational collapse of a class of anisotropic elastic materials. Standard requirements of physical acceptability are satisfied, in particular, existence of an equation of state in closed form, weak energy condition, and existence of a regular Cauchy surface at which the collapse begins. The matter properties are generic in the sense that both the radial and the tangential stresses are non-vanishing, and the kinematical properties are generic as well, since shear, expansion, and acceleration are also non-vanishing. As a test-bed for cosmic censorship, the nature of the future singularity forming at the center is analyzed as an existence problem for o.d.e. at a singular point using techniques based on comparison theorems, and the spectrum of endstates – blackholes or naked singularities – is found in full generality. Consequences of these results on the Cosmic Censorship conjecture are discussed.

The Fermat principle in general relativity and applications

In this paper we use a general version of Fermat’s principle for light rays in general relativity and a curve shortening method to write the Morse relations for light rays joining an event with a smooth timelike curve in a Lorentzian manifold with boundary. The Morse relations are obtained under the most general assumptions and one can apply them to have a mathematical description of the gravitational lens effect in a very general context. Moreover, Morse relations can be used to check if existing models are corrected.

A note on the Morse index theorem for geodesics between submanifolds in semi-Riemannian geometry

The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse index theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse proof. This approach reduces substantially the effort required in the proofs of the theorem given previously [Ann. Math. 73(1), 49–86 (1961); J. Diff. Geom 12, 567–581 (1977); Trans. Am. Math. Soc. 308(1), 341–348 (1988)]. Exactly the same argument works also in the case of timelike geodesics between two submanifolds of a Lorentzian manifold. For the extension to the lightlike Lorentzian case, just minor changes are required and one obtains easily a proof of the focal index theorem previously presented [J. Geom. Phys. 6(4), 657–670 (1989)].

A Fermat Principle on Lorentzian manifolds and applications

Abstract We present a version of the Fermat Principle to Lorentzian manifolds endowed with a time function. The principle is used to obtain some results concerning the existence and multiplicity of light rays, generalizing part of the work in [1–3]. At this time, the results are announced and discussed, while the details of their proofs are left to a forthcoming paper [4].

The Maslov index and a generalized Morse index theorem for non-positive definite metrics

Abstract We present an extension of the celebrated Morse index theorem in Riemannian geometry to the case of geodesics in pseudo-Riemannian manifolds. It is considered the case that both endpoints are variable. The notion of Maslov index for pseudo-Riemannian geodesics replaces the notion of geometric index for Riemannian geodesics.

Multiplicity of solutions to the Yamabe problem on collapsing Riemannian submersions

Let g_t be a family of constant scalar curvature metrics on the total space of a Riemannian submersion obtained by shrinking the fibers of an original metric g, so that the submersion collapses as t approaches 0 (i.e., the total space converges to the base in the Gromov-Hausdorff sense). We prove that, under certain conditions, there are at least 3 unit volume constant scalar curvature metrics in the conformal class [g_t] for infinitely many ts accumulating at 0. This holds, e.g., for homogeneous metrics g_t obtained via Cheeger deformation of homogeneous fibrations with fibers of positive scalar curvature.

A note on the uniqueness of solutions for the Yamabe problem

We prove that in conformal classes of metrics near the class of an Einstein metric (other than the standard round metric on a sphere) the Yamabe problem has a unique solution up to scaling. This is a local extension, in the space of conformal classes, of a well-known uniqueness criterion due to Obata.

On Fermat's principle for causal curves in time oriented Finsler spacetimes

In this work, a version of Fermats principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the second variation of the time arrival functional along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of the critical points of the time arrival functional is investigated and a Morse index theorem in the context of Finsler spacetime is presented.

Lagrangian and Hamiltonian formalism for constrained variational problems

We consider solutions of Lagrangian variational problems with linear constraints on the derivative. These solutions are given by curves