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1: The circle S = { t = 0, r = ¯ r } in the 3-dimensional Minkowski space endowed with the metric ¯ g = −dt 2 + dr 2 + r 2 dϕ 2 admits a focal point along all futurepointing null geodesics with initial velocity k.
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We give a characterization theorem for umbilical spacelike submanifolds of arbitrary di- mension and co-dimension immersed in a semi-Riemannian manifold. Letting the co- dimension arbitrary implies that the submanifold may be umbilical with respect to some subset of normal directions. This leads to the definition of umbilical space and to the study...
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... Introducing this into the lefthand side of condition (3) of Theorem 2.2 and using (6-9) for the Riemann tensor, a little calculation leads to [3] ...
... On a positive side, the condition of the theorems, as given involving quantities of only the extra-dimensional space in comparison with well controllable quantities depending on f and its derivatives, may help in finding the stable possibilities, providing information on which classes of extra-dimensional spaces (Y,ḡ) may be viable and why -and for which warping functions f . Author contributions statement: NC contributed to an earlier version of the present research and performed calculations needed for the material presented in the Appendix as part of her PhD thesis [3]. JMMS conceived and proposed this research, developed the necessary methodology, is accountable for all the steps and calculations of the entire research process, and for structuring and writing the manuscript. ...
... for the general case with arbitrary signatures and dimensions of the fiber and the base. As far as we know these results were first presented in [3]. Let γ : x µ = x µ (u) be any parametrized curve (not necessarily a geodesic) with tangent vector N µ := dx µ (u)/du = (N a ,N i ). ...
New singularity theorems are derived for generic warped-product spacetimes of any dimension. The main purpose is to analyze the stability of (compact or large) extra dimensions against dynamical perturbations. To that end, the base of the warped product is assumed to be our visible 4-dimensional world, while the extra dimensions define the fibers, hence we consider "extra-dimensional evolution". Explicit conditions on the warping function that lead to geodesic incompleteness are given. These conditions can be appropriately rewritten, given a warping function, as restrictions on the intrinsic geometry of the fibers ---i.e. the extra dimensional space. To find the results, the conditions for parallel transportation in warped products in terms of their projections onto the base and the fibers have been solved, a result of independent mathematical interest that have been placed on an Appendix.
