Daniel V. Tausk

Bell's theorem

This chapter discusses the result which has come to be known as ‘Bell’s Theorem’ but which Bell himself instead referred to as the ‘locality inequality theorem’.

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).

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)].

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.

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

Variational aspects of the geodesics problem in sub-Riemannian geometry☆

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Can we make a Bohmian electron reach the speed of light, at least for one instant?

in a differentiable manifold

Compact lines and the Sobczyk property

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Complementary Lagrangians in infinite dimensional symplectic Hilbert spaces.

that are everywhere tangent to a smooth distribution

On the Maslov and the Morse index for constrained variational problems

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