A general class of Lorentzian metrics, $M_0 x R^2$, $ds^2 = <.,.> + 2 du dv + H(x,u) du^2$, with $(M_0, <.,.>$ any Riemannian manifold, is introduced in order to generalize classical exact plane fronted waves. Here, we start a systematic study of their main geodesic properties: geodesic completeness, geodesic connectedness and multiplicity, causal character of connecting geodesics. These results are independent of the possibility of a full integration of geodesic equations. Variational and geometrical techniques are applied systematically. In particular, we prove that the asymptotic behavior of $H(x,u)$ with $x$ at infinity determines many properties of geodesics. Essentially, a subquadratic growth of $H$ ensures geodesic completeness and connectedness, while the critical situation appears when $H(x,u)$ behaves in some direction as $|x|^2$, as in the classical model of exact gravitational wavesComment: Final version with minor errata corrected. 19 pages, Latex. To appear in Gen. Relat. Gravit. (2003

In order to apply variational methods to the action functional for geodesics of a stationary spacetime, some hypotheses, useful to obtain classical Palais-Smale condition, are commonly used: pseudo-coercivity, bounds on certain coefficients of the metric, etc. We prove that these technical assumptions admit a natural interpretation for the conformal structure (causality) of the manifold. As a consequence, any stationary spacetime with a complete timelike Killing vector field and a complete Cauchy hypersurface (thus, globally hyperbolic), is proved to be geodesically connected.

The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes the model problem \ud \[\bar J(u) = \int_\Omega \bar A(x,u)|\nabla u|^p dx - \int_\Omega G(x,u) dx\]\ud in the Banach space $W^{1,p}_0(\Omega)\cap L^\infty(\Omega)$, being $\Omega$ a bounded domain in $\R^N$.\ud In order to use ``classical'' theorems, a suitable variant of condition $(C)$ is proved and $W^{1,p}_0(\Omega)$ is decomposed according to a ``good'' sequence of finite dimensional subspaces

We analyze the extendability of the solutions to a certain second order differential equation on a Riemannian manifold (M, g), which is defined by a general class of forces (both prescribed on M or depending on the velocity). The results include the general time-dependent anholonomic case, and further refinements for autonomous systems or forces derived from a potential are obtained. These extend classical results for Lagrangian and Hamiltonian systems. Several examples show the optimality of the assumptions as well as the utility of the results, including an application to relativistic pp-waves.Key WordsDynamics of classical particles, autonomous and non-autonomous systems, second order differential equation on a Riemannian manifold, completeness of inextensible trajectories.MSC2010. Primary: 34A26, 34C40. Secondary: 37C60, 53D25, 70G45, 83C75.

Geodesics become an essential element of the geometry of a semi-Riemannian manifold. In fact, their differences and similarities with the (positive definite) Riemannian case, constitute the first step to understand semi-Riemannian Geometry. The progress in the last two decades has become impressive, being especially relevant the systematic introduction of (infinite-dimensional) variational methods.Our purpose is to give an overview, from refinements of classical results to updated variational settings. First, several properties (and especially completeness) of geodesics in some ambient spaces are studied. This includes heuristic constructions of compact incomplete examples, geodesics in warped, GRW or stationary spacetimes, properties in surfaces and spaceforms, or problems on stability of completeness.Then, we study the variational framework, and focus on two fundamental problems of this approach, which regards geodesic connectedness. The first one deals with a variational principle for stationary manifolds, and its recent implementation inside Causality Theory. The second one concerns orthogonal splitting manifolds, and a reasonably selfcontained development is provided, collecting some steps spread in the literature.