We discuss the question of whether or not a general Weyl structure is a suitable mathematical model of space-time. This is an issue that has been in debate since Weyl formulated his unified field theory for the first time. We do not present the discussion from the point of view of a particular unification theory, but instead from a more general standpoint, in which the viability of such a structure as a model of space-time is investigated. Our starting point is the well known axiomatic approach to space-time given by Elhers, Pirani and Schild (EPS). In this framework, we carry out an exhaustive analysis of what is required for a consistent definition for proper time and show that such a definition leads to the prediction of the so-called "second clock effect". We take the view that if, based on experience, we were to reject space-time models predicting this effect, this could be incorporated as the last axiom in the EPS approach. Finally, we provide a proof that, in this case, we are led to a Weyl integrable space-time (WIST) as the most general structure that would be suitable to model space-time.

In this paper we show the validity, under certain geometric conditions, of Wheeler's thin sandwich conjecture for higher dimensional theories of gravity. We extend the results shown by R. Bartnik and G. Fodor for the 3-dimensional case in [1] in two ways. On the one hand we show that the results presented in [1] are valid in arbitrary dimensions, and on the other hand we show that the geometric hypotheses needed for the proofs can always be satisfied, which constitutes in itself a new result for the 3-dimensional case. In this way, we show that on any compact n-dimensional manifold, n ≥ 3, there is an open set in the space of all possible initial data where the thin sandwich problem is well-posed.

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In this paper we make a detailed analysis of conservation principles in the context of a family of fourth-order gravitational theories generated via a quadratic Lagrangian. In particular, we focus on the associated notion of energy and start a program related to its study. We also exhibit examples of solutions which provide intuitions about this notion of energy which allows us to interpret it, and introduce several study cases where its analysis seems tractable. Finally, positive energy theorems are presented in restricted situations.

We discuss the possibility of extending different versions of the Campbell-Magaard theorem, which have already been established in the context of semi-Riemannian geometry, to the context of Weyl's geometry. We show that some of the known results can be naturally extended to the new geometric scenario, although new difficulties arise. In pursuit of solving the embedding problem we have obtained some no-go theorems. We also highlight some of the difficulties that appear in the embedding problem, which are typical of the Weylian character of the geometry. The establishing of these new results may be viewed as part of a program that highlights the possible significance of embedding theorems of increasing degrees of generality in the context of modern higher-dimensional space-time theories.