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  • Author: Rui Pacheco x
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In this note we investigate the (E)-index and the (E)-nullity of vacuum solutions from the two-torus into the two sphere.


The isotropy projection establishes a correspondence between curves in the Lorentz–Minkowski space 𝐄13 and families of cycles in the Euclidean plane (i.e., curves in the Laguerre plane 2). In this paper, we shall give necessary and sufficient conditions for two given families of cycles to be related by a (extended) Laguerre transformation in terms of the well known Lorentzian invariants for smooth curves in 𝐄13. We shall discuss the causal character of the second derivative of unit speed spacelike curves in 𝐄13 in terms of the geometry of the corresponding families of oriented circles and their envelopes. Several families of circles whose envelopes are well-known plane curves are investigated and their Laguerre invariants computed.