Abstract
In the present work, we show that in contrast to sub-Riemannian geometry, in sub-Lorentzian geometry the manifold topology, the topology generated by an analogue of the Riemannian distance function and the Alexandrov topology based on causal relations, are not equivalent in general and may possess a variety of relations. We also show that ‘opened causal relations’ are more well-behaved in sub-Lorentzian settings.
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Notes
- 1.
1 Note that we cannot define the eigenvalues of a quadratic form, but their sign due to Sylvester’s Theorem of Inertia. We call the number of negative eigenvalues the index of the form.
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Markina, I., Wojtowytsch, S. (2014). On the Alexandrov Topology of sub-Lorentzian Manifolds. In: Stefani, G., Boscain, U., Gauthier, JP., Sarychev, A., Sigalotti, M. (eds) Geometric Control Theory and Sub-Riemannian Geometry. Springer INdAM Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-319-02132-4_17
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DOI: https://doi.org/10.1007/978-3-319-02132-4_17
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