An introduction to Lorentz surfaces
``This book describes an emerging field of research, and includes detailed proofs of numerous results, many of which have not yet appeared elsewhere.'' Nevertheless, the book is well suited as a text for a special topics course or a seminar for students, too: besides some elementary knowledge in differential geometry, topology, and complex analysis, no qualifications are required to understand the text. The author emphasizes the presentation of the methods and gives elementary proofs where possible; and not only for the few results that are stated without proofs, a comprehensive list of references is given. The book is enriched with a large number of exercises and examples. Lorentz surfaces are the signature analogs of Riemann surfaces: defining two Lorentzian metrics \(h\), \(\widetilde h\), i.e., metrics of signature (1,1), on a (2-dimensional) surface \(S\) to be conformally equivalent if there is a positive function \(\lambda\) on \(S\) with \(\widetilde h= \lambda h\), a Lorentz surface \({\mathcal L}\) is a surface \(S\) equipped with an equivalence class of conformally equivalent metrics \({\mathcal L}= (S,[h])\). Obviously, the causal character \(\text{sign} h(v,v)\) of a tangent vector \(v\) is an invariant of a conformal class \([h]\) of Lorentzian metrics on a surface -- thus, in the theory of Lorentz surfaces the notion of causal character replaces the notion of angles in Riemann surface theory. In fact, the system of null lines on a surface \(S\) determines the corresponding conformal class of Lorentzian metrics as soon as the two systems of null lines are ordered by means of an orientation of \(S\). This is the reason for the prominent position of the study of null line systems in the book: the basic facts are discussed in the first chapter ``Null lines on Lorentz surfaces''. In the second chapter ``Box surfaes, yardsticks and global properties of Lorentzian metrics'', the analog for the familiar interpretation of a Riemann surface as a complex curve is given: equipping the coordinate 2-plane with a Lorentz structure \([dx dy]\), the transition functions of conformal coordinates separate into two functions \(\widetilde x=f(x)\) and \(\widetilde y=g(y)\) since they must preserve the systems of null lines. An atlas of coordinate charts on a surface \(S\) with this form of transition function is called a ``box atlas'' on \(S\) -- and the corresponding ``box surfaces'' are proven to be in one-one correspondence with Lorentz surfaces (Theorem 1). Besides this fundamental result, the existence of ``yardsticks'' -- Riemannian metrics for which the null lines of a Lorentz surface intersect orthogonally -- and of non-vanishing vector fields with a given causal character -- time orientability -- is discussed. Fixing a Lorentzian metric \(h\) on a Lorentz surface \((S,[h])\), the notions of curvature and geodesic lines make sense and, finally, the notion of (null, spacelike, timelike) completeness is introduced. Several examples that illustrate the different types of completeness are given. Any conformal diffeomorphism between Riemann surfaces is automatically analytic. In the case of Lorentz surfaces, the situation is very different: there exist Lorentz surfaces which are \(C^j\) conformally equivalent (i.e., there exists a conformal \(C^j\)-diffeomorphism) but not \(C^{j+1}\) conformally equivalent (Theorem 2). Also, the author gives an infinite family of simply connected Lorentz surfaces that are not conformally equivalent of any order -- which contrasts the uniformization theorem for Riemann surfaces. The second half of the third chapter ``Conformal equivalence and the Poincaré index'' provides a short introduction into the theory of the Poincaré index of a direct field. In particular, Hamburger's method for calculating the Poincaré index is discussed. Applying the Poincaré index to the null direction fields, further results on the behaviour of the system of null lines on a Lorentz surface are obtained, and it is shown that every simply connected Lorentz surface is diffeomorphic to the two-plane. Moreover, relating the sum of the indices of a direction field on a compact surface to its genus, the Poincaré index theorem is used to obtain a more general result: any compact Lorentz surface has genus 1. ``Kulkarni's conformal boundary'' \(\partial {\mathcal L}\) of a Lorentz surface \({\mathcal L}\) is formed from equivalence classes of null rays, that are identified if they lie on the edges of certain types of polygons formed by null lines. The author introduces the conformal boundary, discusses several properties and defines a topology on \({\mathcal L} \cup \partial {\mathcal L}\) that is slightly different from Kulkarni's. In Smyth's theorem a first method is given how to identify a Lorentz surface in terms of its conformal boundary. ``Using the conformal boundary'': in the following two chapters, further classification results for Lorentz surfaces are obtained. In particular, the question when a (simply connected) Lorentz surface can be realized as a subset of the Minkowski plane \((E^2, [dx dy])\) is attacked (Theorem of Luo-Stong). Here, the behaviour of the systems of null curves play a crucial role -- whether all the null curves passing through a given null curve (the span of the given null curve) cover the Lorentz surface (Lemma 25) and whether a null curve can continuously be mapped into the conformal boundary (Theorem 6). In Chapter 6, ``Conformal invariants on Lorentz surfaces'', conformal indices of a Lorentz surface or its conformal boundary are introduced: for example, the number of null lines needed to join two points, or the number of distinct spans of null lines can be used to define conformal indices for a Lorentz surface. Throughout the last two chapters, the geometry of immersed surfaces in 3-space is discussed: in Chapter 7, ``Classical surface theory and harmonically immersed surfaces'', the theories of surfaces in Euclidean 3-space and of space- or timelike surfaces in Minkowski 3-space are contrasted. The Hilbert-Holmgren theorem for harmonic maps prepares the study of timelike minimal surfaces in the last Chapter 8, ``Conformal realization of Lorentz surfaces in Minkowski 3-space'': if the metric induced by a timelike minimal surface in Minkowski space as the pull back of the canonical Euclidean metric on the ambient space is complete, then the universal cover of the minimal surface is conformally diffeomorphic to the Minkowski plane \(E^{2,1}\). And, if a timelike minimal surface is entire, i.e., graph over a plane, then it is conformally equivalent to the Minkowski plane (conformal Bernstein type theorem). Using a Weierstrass representation for timelike minimal immersions into Minkowski 3-space, the associated family of timelike minimal surfaces is examined and several examples of embeddings of Lorentz surfaces as timelike (minimal) surfaces into Minkowski space are discussed. Finally, returning to the question of Chapter 5, it is stated that if a simply connected timelike surface is complete as a surface in Euclidean space, then it is conformally diffeomorphic to a subset of the Minkowski plane (Lin's theorem). According to the reviewer's opinion, the author certainly succeeds in her goal: ``The text outlines much of what is known about Lorentz surfaces. The book is written to make an emerging field accessible to graduate students and professionals alike''.
scientific article; zbMATH DE number 917504