Prof. Dr. Helga Baum
Publikationsliste

1.  Einige Anwendungen der Ganzzahligkeitssätze für charakteristische Klassen.
Diplomarbeit, Humboldt-Universität Berlin, 1977

2. (with T. Friedrich): Immersionen höherer Ordnung kompakter Mannigfaltigkeiten in Euklidische Räume.
Beiträge zur Algebra und Geometrie, 9 , 1980, 83-101

3.  (with T. Friedrich): Spektraleigenschaften des Dirac-Operators. Die Fundamentallösung seiner Wärmeleitungsgleichung und die Asymptotenentwicklung seiner Zeta-Funktion.
Journal of Differential Geometry, 15, 1980, 1-26

4. (with T. Friedrich): Spectral Properties of the Dirac operator.
Bulletin d'la Academie d. Science Polonaise, Vol. XXVII, Nr. 7-8, 1979, 621-624

5.  Selfdual connections and holomorphic bundles.
In: Riemannian Geometry and Instantons, Teubner-Text zur Mathematik, Band 34,  105-118, Teubner-Verlag Leipzig, 1981

6.  Spin-Strukturen und Dirac-Operatoren über pseudo-Riemannschen Mannigfaltigkeiten.
Dissertation A, Humboldt-Universität Berlin, 1980

7.  Spin-Strukturen und Dirac-Operatoren über pseudo-Riemannschen Mannigfaltigkeiten.
Teubner-Text zur Mathematik, Band 41, Teubner-Verlag Leipzig, 1981

Abstract: By way of a generalization of the classical Dirac equation of relativistic quantum mechanics, this book deals with Dirac operators on pseudo-Riemannian manifolds. Chapter one introduces the necessary algebraic material, while chapter two discusses spinor structures on pseudo-Riemannian manifolds, the existence of which is essential for the definition of the Dirac operators. The third chapter is devoted to the geometrical, analytical and spectral properties of the Dirac operator, and finally chapter four is concerned with the existence of parallel spinors on pseudo-Riemannian manifolds.

8.  Spinor structures and Dirac operators on pseudo-Riemannian manifolds.
Bulletin d'la Academie d. Science Polonaise, Vol. 32, Nr. 3-4, 1985, 155-171.
Kurzfassung in: Proceedings of the Conference on Differential Geometry and its Applications, Novo Mesto 1980, 17-23.

9.  The index of the pseudo-Riemannian Dirac operator as a transversally elliptic operator.
Annals of Global Analysis and Geometry, 11(2), 1983, 11-20

10.  1-forms on the moduli space of irreducible connections defined by the spectrum of Dirac operators.
Journal of Geometry and Physics, Vol. IV, No. 4, 1987, 503-521

Abstract: We study 1-forms on the moduli space of irreducible connections of a G-principal bundle P over a closed Riemannian spin manifold M which are defined by the coefficients of the asymptotic expansion of the trace of a certain operator. In particular, we obtain a foliation of codimension five of the space of G-instantons of the sphere S4 .

11.  Odd-dimensional Riemannian manifolds with immaginary Killing spinors.
Annals of Global Analysis and Geometry, 7(2), 1989, 141-15

12.  Complete Riemannian manifolds with imaginary Killing spinors.
Annals of Global Analysis and Geometry, 7(3), 1989, 205-226

Abstract: We describe the structure of all complete connected Riemannian spin manifolds with imaginary Killing spinors. We prove that such a manifolds is a warped product of a complete connected spin manifold with non-zero parallel spinor fields and the real axis

13.  Varietes riemanniennes admettant des spineurs de Killing imaginaires.
C.R.Acad. Sci. Paris, t.309, Serie I, 1989, 47-47

Das ist die Kurzfassung der Ergebnisse der beiden vorstehenden Artikel

14.  Vollständige, nicht-kompakte Mannigfaltigkeiten mit Killing-Spinoren und Spektralinvarianten des Dirac-Operators als Funktionen auf dem Moduli-Raum der Eichfeldtheorie.
Dissertation B, Humboldt-Universität Berlin, 1989

15.  Killing spinors on Riemannian manifolds.
Vortrag auf der Arbeitstagung in Bonn, 1990, MPI-Preprint 90-52

16.  Une borne superieure pour la premiere valeur de l'operateur de Dirac sur une variete riemannienne spinorielle compacte a courbure scalaire positive.
C.R. Acad. Sci. Paris, t.311, Serie I, 1990, 389-392

17.  An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds.
Mathematische Zeitschrift, 206(3), 1991, 409-422

Abstract:  By comparison with the sphere and perturbation methods Vafa and Witten proved the existence of  universal upper bounds for the eigenvalues of the twisted Dirac operator on compact manifolds (CMP 95, 1984). Using this method we derive a geometric upper bound for the smallest eigenvalue of the classical Dirac operator.

18.  (with T.Friedrich, I.Kath, R.Grunewald): Twistors and Killing Spinors on Riemannian Manifolds.
Teubner-Texte zur Mathematik, Band 124, Teubner-Verlag Stuttgart/Leipzig 1991

Abstract: In this book we investigate, after an introductory section to Clifford algebras, spinors on manifolds etc., in particular solutions of the twistor equation as well as Killing spinors. New results on the construction and classification of Riemannian manifolds with real and imaginary Killing spinors, respectively, are the main subject of this book. Moreover, we consider the relations between solutions of the general twistor equation and Killing spinors.

19.  The Zeta-invariant of  Dirac operators coupled to instantons.
SFB288- preprint No. 90, 1993,  [dvi] , [ps]

Abstract: We calculate the Zeta-invariant of the square of  the Dirac operator coupled to instantons

20.  Eigenvalue estimates for Dirac operator coupled to instantons.
A final version appeared in: Annals of Global Analysis and Geometry, 12 (1994), 193-209,  [dvi], [ps]

Abstract: We prove a sharp lower bound for the first positive eigenvalue of Dirac operators coupled to instantons and discuss the limit case.

21.  A remark on the spectrum of the Dirac operator on pseudo-Riemannian spin manifolds.
SFB288- preprint No. 136, 1994,  [dvi], [ps]

Abstract: It is proved that the eigenvalues of the Dirac operator on even dimensional space- and time-oriented pseudo-Riemannian spin manifolds lie symmetric to the real and to the immaginary axes, and that the Dirac operator does not have residual spectrum if the associated Riemannian metric is complete.

22.  (with T. Friedrich): Eigenvalues of the Dirac operator, Twistors and Killing spinors on Riemannian manifolds.
in: Clifford Algebras and Spinor Structures,  ed. by R.Ablamowicz and R.Lounesto, Kluwer Academic Publishers 1995, 243-256.  [dvi], [ps]

23.  (with  I. Kath): Normally hyperbolic operators, the Huygens property and conformal geometry.
(In memory of P. Günther)
A final version appeared in: Annals of Global Analysis and Geometry 14, 1996, 315-371  [dvi], [ps]

Abstract: In this paper we give a review on normally hyperbolic operators of Huygens type. The method to determine Huygens operators we explain here were essentially influenced and developed by P. Günther.

24.  The Dirac operator on Lorentzian spin manifolds and the Huygens property.
A final version appeared in: Journal of Geometry and Physics 23, 1997, 42-64.  [dvi], [ps]

Abstract: We consider the Dirac operator D of a Lorentzian spin manifold of even dimension greater or equal four. We prove that the square D2 of the  Dirac operator on plane wave manifolds and the shifted operator D2-K on Lorentzian space forms of constant sectioneal curvature K are of Huygens type. Furthermore, we study the Huygens property for coupled Dirac operators on four-dimensional Lorentzian spin manifolds.

25.  Strictly pseudoconvex spin manifolds, Fefferman spaces and Lorentzian twistor spinors.
SFB288-preprint No. 250, 1997  [dvi], [ps]

Abstract: We prove that there exist global solutions of the twistor equation on the (modified) Fefferman spaces of strictly pseudoconvex spin manifolds of arbitrary dimension and we study their properties.

26.  Lorentzian twistor spinors and CR-geometry.
A final version appeared in: Differential Geometry and its Applications, Vol. 11(1999), No 1, S. 69-96.  [dvi], [ps]

This is the shortened version of the SFB288-preprint No. 250, 1997

27.  (with I.Kath): Parallel spinors and holonomy groups on pseudo-Riemannian spin manifolds.
SFB288- preprint No. 276, 1997  [dvi], [ps]

Abstract: We describe the possible holonomy groups of simply connected irreducible non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel spinors.  Furthermore, we determine the chiral and the causal type of the  parallel spinors. (1. version of the proof)

28.  (with I.Kath): Parallel spinors and holonomy groups on pseudo-Riemannian spin manifolds.
A final version appeared in:  Annals of Global Analysis and Geometry, 17 (1), 1-17, 1999.  [dvi], [ps]

Abstract: We describe the possible holonomy groups of simply connected irreducible non-locally symmetric pseudo-Riemannian spin manifolds which admit parallel spinors. Furthermore, we determine the chiral and the causal type of the  parallel spinors. (2. version of the proof)

29.  Twistor spinors on Lorentzian manifolds, CR-geometry and Fefferman spaces.
Proceedings of the 7. International Conference on Differential Geometry and Applications, Brno, 10.-14.8.1998, Brno 1999, 29-38.  [dvi], [ps]

Abstract: The lecture deals with twistor spinors on Lorentzian manifolds. In particular, we explain a relation between a certain class of Lorentzian twistor spinors and the Fefferman spaces of strictly pseudoconvex spin manifolds which appear in CR-geometry.

30.  Twistor spinors on Lorentzian symmetric spaces.
 Journal of Geometry and Physics 34  (2000), 270-286   [dvi], [ps]

Abstract: An indecomposable (= irreducible) Riemannian symmetric space which admits non-trivial twistor spinors has constant sectional curvature.  Furthermore, each homogeneous Riemannian manifold with parallel spinors is flat. In this paper we solve the twistor equation on all indecomposible Lorentzian symmetric spaces explicitly. In particular, we show, that there are indecomposable Lorentzian symmetric spaces (those with solvable transvection group) which  admits twistor spinors and are not of constant curvature. Furthermore, this gives examples of non-flat and non-Ricci-flat homogeneous Lorentzian manifolds with parallel spinors.

31.  Twistor and Killing spinors in Lorentzian Geometry.
 Seminaires et Congres 4, Societe Mathematique de France, 2000, 35-52   [dvi], [ps]

Abstract: This paper is a survey about recent results concerning twistor and Killing spinors on Lorentzian manifolds based on lectures given at CIRM, Luminy, in June 1999, and at ESI, Wien, in October 1999. After some basic facts about twistor spinors we explain a relation between Lorentzian twistor spinors with lightlike Dirac current  and the Fefferman spaces of stricly pseudoconvex spin manifolds which appear in CR-geometry. Secondly, we discuss the relation between  twistor spinors with timelike Dirac current and Lorentzian Einstein-Sasaki structures. Then, we indicate the local structure of all Lorentzian manifolds carrying real Killing spinors. In particular, we show a global Splitting Theorem for complete Lorentzian manifolds in the presence of Killing spinors. Finally, we review some facts about parallel spinors in Lorentzian geometry.

32.  Conformal Killing spinors and special geometric structures in Lorentzian geometry - a survey.  [dvi]  ,  [ps]
Proceedings of the Workshop on Special Geometric Structures in String Theory, Bonn, September 2001. 
Proceedings archive of the EMS Electronic Library of Mathematics, www.univie.ac.at/EMIS/proceedings/

Abstract:  This paper is a survey on special geometric structures that admit conformal Killing spinors based on lectures given at the "Workshop on Special Geometric Structures in String Theory", Bonn, September 2001,  and during the program on String Theory at the Ernst Schrödinger Institut, Wien , November 2001.

 
33.   CR-Geometry and Conformally Invariant Spinor field Equations
In:  Selected Topics from Cauchy-Riemann Geometry.  Ed. by Sorin Dragomir.  Quaderni di mathematica, Vol. 9,  41-87, 2002.
  

34.  (with I. Kath):  Doubly extended Lie Groups - Curvature, Holonomy and Parallel Spinors.
Diff. Geom. and its Appl. 19 (2003), 253 - 280.  [dvi]  [ps]

Abstract:  In the present article we study the geometry of doubly extended Lie groups with their natural biinvariant metric. We describe the curvature, the holonomy and the space of parallel spinors. This is completely done for all simply connected groups with biinvariant metric of Lorentzian signature (1,n-1), of signature (2,n-2)  and of signature (p,q), where p+q<7.  Furthermore, some special series with higher signature are discussed. In particular, this paper contains the  (new)  classification of all metric Lie groups up to dimension 6.

 
35. (with F. Leitner): The twistor equation in Lorentzian spin geometry.
Mathematische Zeitschrift  247 (4) (2004), 795 - 812.  [dvi] [ps]

Abstract:  In this paper we discuss the twistor equation in Lorentzian spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds, which admit twistor spinors inducing lightlike Dirac currents. Furthermore, we derive all local geometries with singularity free twistor spinors that occur up to dimension 7.

36.  (with Felipe Leitner): The geometric structure of Lorentzian manifolds with twistor spinors in low dimension.
In: Dirac Operators - Yesterday and Today. eds: J.P. Bourguignon, T. Bransen, A. Chamseddine, O. Hijazi, R. Stanton;  229- 240, International Press 2005   [dvi], [ps]
 
Abstract: In this paper we discuss the twistor equation in Lorentzain spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds , which admit twistor spinors inducing lightlike Killing fields. Furthermore,  we classify (up to local conformal transformations) all geometries with singularity free twistor spinors that occur up to dimension 6. An extended version and complete proofs can be found in the paper 35

 
37. (with Olaf Müller): Codazzi spinors and globally hyperbolic Lorentzian manifolds with special holonomy.
Mathematische Zeitschrift  258 (2008), 185-211.  [pdf]
 
Abstract: In this paper we prove that any Lorentzian holonomy group of the form G xs Rn-2 , where G is trivial or a product of groups SU(k), Sp(l), G2 or Spin(7) can be realized by a globally hyperbolic Lorentzian manifold with geodesically complete Cauchy surfaces.  For that aim we describe the structure of Riemannian manifolds with Codazzi spinors to invertible Codazzi tensors. 

 
38. Conformal Killing spinors and the holonomy probelm in Lorentzian geometry - a survey of new results.
In: Symmetries and Overdetermined Systems of Partial Differential Equations, eds. M. Eastwood, W. Miller, 251--264, IMA Volumes in Mathematics, Springer 2008.   [ps]   [pdf]
  
Abstract: This paper is a survey of recent results about conformal Killing spinors in Lorentzian geometry based on a lecture given during the Summer Program  Symmetries and Overdetermined Systems of Partial Differential Equations at IMA, Minnesota, 17.07.06 -04.08.06. In particular, we will focus on a special class of geometries admitting conformal Killing spinors - on Brinkmann spaces with parallel spinors. We will discuss their holonomy groups and the global realizability as globally hyperbolic spaces.
  
 
39. D. Alekseevsky, H. Baum (eds): Recent Developments in Pseudo-Riemannian Geometry
ESI Lecture Series in Mathematics and Physics, EMS Publishing House 2008, 537 pp   Flyer
  
 
40. Helga Baum: Eichfeldtheorie.  Eine Einführung in dei Differenmtialgeometrie auf Faserbündeln.
Springer-Verlag 2009.  Flyer
  
 
41. Helga Baum and Andreas Juhl: Conformal Differential Geometry. Q-curvatrue and conformal holonomy.
Oberwolfach Seminars, Vol. 40, Birkhäuser-Verlag,  2010.   Flyer
  
 
42. The conformal analog of Calabi-Yau manifolds.  [ps]   [pdf]
In:  Handbook of Pseudo-Riemannian Geometry and Supersymmerty, IRMA Lectures in Mathematics and Theoretical Physics, Vol: 16, eds. V. Cortés, Publishing House of the EMS, 2010
  
Abstract: This survey intends to introduce the reader to holonomy theory of Cartan connections. Special attention is given to the normal conformal Cartan connection, uniquely defined for a class of  conformally equivalent metrics, and to its holonomy group - the 'conformal holonomy group'. We explain the relation between conformal holonomy group and existence of Einstein metrics in the conformal class as well as the relation between conformal holonomy group and existence of conformal Killing spinors. In particular, we describe Lorentzian manifolds with conformal holonomy group in SU(1,m), which can be viewed as conformal analog of Calabi-Yau manifolds. Such Lorentzian metrics, known as Fefferman metrics, appear on S1-bundles over strictly pseudoconvex CR spin manifolds.
  
43. Holonomy groups of Lorentzian manifolds - a status report.  [pdf]
In: Global Differential Geometry, eds. C.Bär, J. Lohkamp and M. Schwarz,. p.163-200, Springer Proceedings in Mathematics Vol. 17, Springer-Verlag,  2012.
 
44. (with Kordian Lärz and Thomas Leistner): On the full holonomy group of special Lorentzian manifolds. 
Mathematische Zeitschrift 277 (2014), 797-828 
 
Abstract: We study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. We prove several results that are based on the classification of the restrictad holonomy groups of such manifolds and provide a construction method  for manifolds with disconnected holonomy which starts from a Riemannian manifold and a properly discontinouous group of isometries. Most of our examples are quotients of pp-waves with disconnected holonomy and without parallel spinor field. Furthermore, we classify the full holonomy group of solvable Lorentzian symmetric spaces and of Lorentzian manifolds with parallel spinor. Finally, we construct examples of globaly hyperbolic manifolds with complete spacelike Cauchy surfaces, disconnected full holonomy and parallel spinor. 
45. Helga Baum: Eichfeldtheorie.  Eine Einführung in die Differentialgeometrie auf Faserbündeln.
2. Auflage, Springer-Verlag 2014

46. Helga Baum, Thomas Leistner, and Andree Lischewski: Cauchy problems for Lorentzian manifolds with special holonomy.  [pdf]
Differential Geometry and its Applications 45 (2016), 43-66

Abstract:  On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions can be extended to a Lorentzian manifold with a parallel null vector field. Similarly, every parallel null spinor on a Lorentzian manifold induces an imaginary generalised Killing spinor on a space-like hypersurface. Then, based on the fact that a parallel spinor field induces a parallel vector field, we can apply the first result to prove: every real analytic Riemannian manifold carrying a real analytic, imaginary generalised Killing spinor can be extended to a Lorentzian manifold with a parallel null spinor, where we can prescribe the form of the ambient metric. Finally, we give examples of geodesically complete Riemannian manifolds satisfying the constraint conditions.

47. Helga Baum: Lorentzian manifolds with special holonomy - Constructions and global properties.
In: SPACE - TIME - MATTER. Analytic and Geometric structures. Eds. J. Brüning and Matthais Staudacher. p. 51-68, De Gruyter Verlag, 2018.

48. Helga Baum and Thomas Leistner: Lorentzian Geometry: Holonomy, Spinors, and Cauchy Problems.
In:  Geometric Flows and the Geometry of Space-time, edited by Vicente Cortés, Klaus Kröncke and Jan Louis, vol 2 in Tutorials, Schools, and Workshops in the Mathematical Sciences, Birkhäuser/Springer, 2018, pp 1-76, publisher link.