Neutral geometry and the Gauss-Bonnet theorem for two-dimensional pseudo- Riemannian manifolds (Q1802702)

The author proves an analogue of the classical Gauß-Bonnet theorem for domains with piecewise smooth boundary in 2-dimensional Lorentz manifolds. This is a generalization of a Gauß-Bonnet formula for the special case that all boundary curves are timelike which has been given by \textit{G. S. Birman} and \textit{K. Nomizu} [Mich. Math. J. 31, 77-81 (1984; Zbl 0591.53053)]. For this generalization he has to define angles between spacelike and timelike vectors. This is done using a linear ``anti-isometry'' which interchanges the timelike and the spacelike vectors at a given point. (This transformation is given in terms of any orthonormal basis and interchanges these basis vectors. Since it commutes with \(SO(1,1)\) it is well defined.) The angle between vectors of different causal character (where one has to invoke the anti-isometry) is complex. With the author's definition of angle the proof of the generalized Gauß-Bonnet theorem is a straightforward modificaton of Birman and Nomizu's proof. Formally his Gauß-Bonnet formula can be obtained from the Riemannian Gauß-Bonnet formula by replacing the summand \(2\pi\) with \(2\pi i\).