Wolfgang Kühnel

SELECTED TOPICS OF MY RESEARCH IN DIFFERENTIAL GEOMETRY AT PRESENT AND IN THE PAST:

Total curvature of smooth submanifolds of Euclidean space

The classical Gauss-Bonnet formula holds for compact even-dimensional manifolds in the extrinsic and an intrinsic form. In odd dimensions we have no satisfying analogue. It does not even seem to be completely clear what values the total curvature of a compact odd-dimensional hypersurface can attain and how one can get representatives for each possible value.. For complete but non-compact submanifolds the situation is even more difficult, see

• F. Dillen and W. Kühnel, Total curvature of complete submanifolds of Eⁿ, Tôhoku Math. J. 57, 171-200 (2005) (pdf)

Total absolute curvature and tightness of smooth submanifolds of Euclidean space

Tightness of smooth submanifolds refers to attaining the minimum total absolute curvature, for a survey and a bibliography see

• T.F. Banchoff and W. Kühnel, Tight submanifolds, smooth and polyhedral, in: Tight and taut submanifolds (T.E. Cecil and S.-s. Chern, eds.), MSRI Publications Vol. 32, 51-118, Cambridge University Press 1997
• T. E. Cecil and W. Kühnel, Bibliography on tight, taut and isoparametric submanifolds, ibid. 307-339

The case of tight surfaces with boundary in 3-space was investigated in

• W. Kühnel and G. Solanes, Tight surfaces with boundary, Bulletin London Math. Soc. 43, 151-163 (2011)

For an extension to the case of non-compact submanifold and the case of a compact space-form as ambient space see the following work of my former students

• M. van Gemmeren, Total absolute curvature and tightness of noncompact manifolds, Transactions of the American Mathematical Society 348, 2413-2426 (1996)
• M.-O. Otto, Tight surfaces in three-dimensional compact Euclidean space forms, Transactions of the American Mathematical Society 355, 4847-4863 (2003)

Weingarten surfaces

By definition a Weingarten surface satisfies a certain relation between the principal curvatures,see

• F. Dillen and W. Kühnel, Ruled Weingarten surfaces in Minkowski 3-space, manuscripta math. 98, 307-320 (1999)
• W. Kühnel and M. Steller, On closed Weingarten surfaces, Monatsh. Math. 146, 113-126 (2005) (pdf )

Conformal geometry of semi-Riemannian manifolds

Conformal mappings and conformal vector fields were intensively studied in both Riemannian and pseudo-Riemannian geometry. Conformally flat spaces have been characterized by Cotton, Finzi and Schouten in the early 20th century. In General Relativity conformal aspects are of importance. For global conformal geometry, the conformal development map was introduced by Kuiper in 1949, after earlier work by Brinkmann in the 1920's. Essential conformal vector fields on Riemannian spaces have been studied by Obata, Lelong-Ferrand and Alekseevskii. Conformal gradient fields are essentially solutions of the differential equation $\; \nabla ^2\varphi = \frac{\Delta \varphi}{n} \cdot g\;$. This equation has been studied since the 1920's by Brinkmann, Fialkow, Yano, Obata, Kerbrat and others. In the Riemannian case the results are quite complete. In the pseudo-Riemannian case we started a systematic approach including a conformal classification theorem in the papers:

• W. Kühnel and H.-B. Rademacher, Essential conformal fields in pseudo-Riemannian geometry, J. Math. Pures et Appl.(9) 74, 453--481 (1995); Part II in J. Math. Sci. Univ. Tokyo 4, 649--662 (1997)
• W. Kühnel and H.-B. Rademacher, Liouville's theorem in conformal geometry, J. Math. Pures et Appl. (9) 88, 251-260 (2007)
• W. Kühnel and H.-B. Rademacher, Conformal transformations of pseudo-Riemannian manifolds,
  in: Recent developments in pseudo-Riemannian geometry (D.Alekseevsky and H.Baum, eds.),
  ESI Lectures in Mathematics and Physics, 261-298, European Math. Society 2008
• W. Kühnel and H.-B. Rademacher, Einstein spaces with a conformal group, Results Math. 56, 421-444 (2009)

Conformal vector fields on space-times

• W. Kühnel and H.-B. Rademacher, Conformal Ricci collineations of space-times,
  Gen. Relativity and Gravitation 33, 1905-1914 (2001)
• W. Kühnel and H.-B. Rademacher, Conformal geometry of gravitational plane waves,
  Geometriae Dedicata 109, 175-188 (2004)

Twistor spinors

Twistor spinors are conformally invariant. Moreover, a twistor spinor induces a conformal vector field. Therefore, by simular methods one can classify spaces carrying twistor spinors.

• W. Kühnel and H.-B. Rademacher, Twistor spinors with zeros, Intern. J. Math. 5, 877--895 (1994)
• W. Kühnel and H.-B. Rademacher, Twistor spinors and gravitational instantons, Lett. Math. Phys. 38, 411--419 (1996)
• W. Kühnel and H.-B. Rademacher, Twistor spinors on conformally flat manifolds, Illinois J. Math. 41, 495--503 (1997)
• W. Kühnel and H.-B. Rademacher, Asymptotically Euclidean manifolds and twistor spinors,
  Commun. Math. Phys. 196, 67--76 (1998), Corr. ibid. 207, 735 (1999)

Triangulations with few vertices

Determine the minimum number of vertices for a triangulation of a given manifold, or find "good" triangulations with a reasonable number of vertices and/or with high symmetry. Find general lower bounds for the possible numbers of vertices. A basic inequality was given in

• U. Brehm and W. Kühnel, Combinatorial manifolds with few vertices, Topology 26, 465-473 (1987).

Upper and Lower bound conjectures for combinatorial manifolds

There is the following CONJECTURE: For any 2k-dimensional combinatorial manifold M with n vertices the following inequality holds: ${{n-k-2} \choose {k+1}} \geq (-1)^k {{2k+1} \choose {k+1}}(\chi(M) - 2)$ with equality if and only if the triangulation is $(k+1)$-neighborly. For a first approach toward this conjecture see

• I. Novik, Upper bound theorems for homology manifolds, Israel J. Math. 108, 45-82 (1998)
• I. Novik, On face numbers of manifolds with symmetry, Advances Math. 192, 183-208 (2005)

For the positive answer to the conjecture see

• I. Novik and Ed Swartz, Socles of Buchsbaum modules, complexes and posets, Advances Math. 222, 2059-2084 (2009).

A similar conjecture for the case of centrally-symmetric triangulations were first studied in the following work by my former student

• E. Sparla, An upper and a lower bound theorem for combinatorial 4-manifolds, Discrete Comp. Geom. 19, 575-593 (1998)
• E. Sparla, A new lower bound theorem for combinatorial 2k-manifolds, Graphs and Combinatorics 15, 109-125 (1999)

Tight polyhedral embeddings

There is the following conjecture: For any tight polyhedral embedding of a (k-1)-connected 2k-dimensional manifold M into Euclidean d-space (not in any hyperplane) the following inequality holds: ${{d-k-1} \choose {k+1}} \leq (-1)^k {{2k+1} \choose {k+1}}(\chi(M) - 2)$ with equality if and only if the image is a (k-1)-neighborly subcomplex of the d-dimensional simplex. For partial results see

• W. Kühnel, Tight polyhedral submanifolds and tight triangulations, Lecture Notes in mathematics 1612, 122 pages, Springer 1995

More recently new tight polyhedral embeddings of 4-manifolds were investigated, see

• M. Casella and W. Kühnel, A triangulated K3 surface with the minimum number of vertices, Topology 40, 753-772 (2001)
• W. Kühnel and F. H. Lutz, A census of tight triangulations, Periodica Math. Hungarica 39, 161-183 (1999)
• W. Kühnel, Tight embeddings of simply connected 4-manifolds, Documenta Math. 9, 401-412 (2004) (electronic)

For tight polyhedral submanifolds of higher-dimensional octahedra see

• W. Kühnel, Centrally-symmetric tight surfaces and graph embeddings, Beiträge zur Algebra und Geometrie 37, 347-354 (1996)
• F. Effenberger and W. Kühnel, Hamiltonian submanifolds of regular polytopes, Discrete Comput. Geometry 43, 242-262 (2010)

There is a close connection with the subject "Upper and lower bound theorem", because k-Hamiltonian 2k-manifolds in the boundary complex of a cross polytope belong to both categories, at least in the centrally-symmetric case. Further progress was made in

• F. Effenberger, Stacked polytopes and tight triangulations of manifolds, J. Comb. Theory, Ser. A 118, No. 6, 1843-1862 (2011)
• F. Effenberger, Hamiltonian submanifolds of regular polytopes, Doctoral Thesis, Stuttgart 2010

Combinatorial manifolds satisfying complementarity, and triangulated projective planes

The problem arises from three different points of view:

• Find triangulations of 2k-manifolds with 3k+3 vertices. This is the minimum number for any manifold which is not a sphere, see
  • U. Brehm and W. Kühnel, Combinatorial manifolds with few vertices, Topology 26, 465-473 (1987)
• Find tight polyhedral embeddings of the projective planes over the real, complex numbers, quaternions, or octonions in the same codimension as the classical Veronese-type embeddings of them. Halfway between two antipodal copies of them we find polyhedral analogous of the Cartan isoparametric hypersurfaces in spheres (which are tubes around the Veronese-type embeddings), see
  • T.F. Banchoff and W. Kühnel, Tight polyhedral models of isoparametric families, and PL--taut submanifolds,
    Advances in Geometry 7, 613--639 (2007)
• Find triangulated (pseudo-)manifolds satisfying the following combinatorial complementarity condition:
  • A subset $A \subset V$ of the set of vertices V spans a simplex if and only if the complement $V \setminus A$ does not.
  This can be regarded as a kind of a combinatorial duality condition. Among other implications, it leads to a triangulated sphere as the deleted join (the so-called Bier-sphere after work of Thomas Bier).

So far the only known examples are the 6-vertex RP², the 9-vertex CP² and a number of 15-vertex triangulations of an 8-manifold, presumably HP². The case of a 27-vertex triangulation of the octonion plane is still open, see

• U. Brehm and W. Kühnel, 15-vertex triangulations of an 8-manifold, Math. Annalen 294, 167-193 (1992)

Triangulations of the d-dimensional torus

There is the following conjecture: Any combinatorial triangulation of the d-dimensional torus must have at least 2d+3 vertices. Examples attaining this bound are available, see

• W. Kühnel and G. Lassmann, Combinatorial d-tori with a large symmetry group, Discrete Comp. Geom. 3, 169-176 (1988)
• G. Dartois and A. Grigis, Separating maps of the lattice E₈ and triangulations of the eight-dimensional torus, Discrete Comp. Geom. 23, 555-567 (2000)

In the case of lattice triangulations the conjecture is proved in

• U. Brehm and W. Kühnel, Lattice triangulations of E³ and of the 3-torus , Israel J. Math. 189, 97-133 (2012).

Manifolds associated with abstract regular polytopes

Abstract regular polytopes are natural generalizations of the Platonic solids and the other classical regular polytopes. The requirement is that the automorphism group acts transitively on flags. So in particular one can transform any point to any other and then, while keeping the point fixed, any edge to any other and so on. For a monograph on the subject see

• P. McMullen and E. Schulte, Abstract Regular Polytopes, 570 pages, Cambridge University Press 2001

Among the abstract regular polytopes there are 4-polytopes with toroidal facets. When transforming these abstract toroidal facets into solid tori, a 3-manifold can be associated with the polytope, see

• U. Brehm, W. Kühnel and E. Schulte, Manifold structures on abstract regular polytopes, Aequationes math. 49, 12-35 (1995)

The higher-dimensional case is still work in progress. 

Publikationsliste

Selected publications by our research group (differential geometry and topology)

Textbooks:

• W. Kühnel, Matrizen und Lie-Gruppen, eine geometrische Einführung, Vieweg+Teubner 2011
• W. Kühnel, Differentialgeometrie, Kurven - Flächen - Mannigfaltigkeiten, Vieweg 1999,
  2. Auflage 2003, 3. Auflage 2005, 4. Auflage 2008, 5. Auflage Vieweg+Teubner 2010, 6. Auflage Springer Spektrum 2013
  English version: Differential Geometry, Curves - Surfaces - Manifolds, translated from the German by B.Hunt,
  AMS Student Mathematical Library Series Vol. 16, American Mathematical Society 2002, 2nd edition 2006

Survey Articles, Monographs:

• W. Kühnel and H.-B. Rademacher, Conformal transformations of pseudo-Riemannian manifolds, in:
  Recent developments in pseudo-Riemannian geometry (D.Alekseevsky and H.Baum, eds.),
  ESI Lectures in Mathematics and Physics, 261-298, European Math. Society 2008
• T.F. Banchoff and W. Kühnel, Tight submanifolds, smooth and polyhedral, in: Tight and taut submanifolds
  (T.E. Cecil and S.-s. Chern, eds.), MSRI Publications Vol. 32, pp. 51-118, Cambridge University Press 1997
• T. E. Cecil and W. Kühnel, Bibliography on tight, taut and isoparametric submanifolds, ibid. pp. 307-339
• W. Kühnel, Tight polyhedral submanifolds and tight triangulations, 122 p., Lecture Notes in Mathematics
  Vol. 1612, Springer 1995
• W. Kühnel, Triangulations of manifolds with few vertices, Advances in Differential Geometry and Topology
  (F. Tricerri, ed.), pp. 59-114, Proceedings of a workshop at the Institute for Scientific Interchange (Torino, Italy),
  World Scientific Publ. 1990

On the Differential Geometry of Submanifolds:

• W. Kühnel and G. Solanes, Tight surfaces with boundary, Bulletin London Math. Soc. 43, 151-163 (2011)
• M. Steller, A Gauss-Bonnet formula for metrics with varying signature, Z. Anal. Anw. 25, 143-162 (2006)
• W. Kühnel and M. Steller, On closed Weingarten surfaces, Monatsh. Math. 146, 113-126 (2005)
• F. Dillen and W. Kühnel, Total curvature of complete submanifolds of Euclidean space,
  Tôhoku Math. J. 57, 171-200 (2005)
• M.-O. Otto, Tight surfaces in three-dimensional compact Euclidean space forms,
  Transactions Amer. Math. Soc. 355, 4847-4863 (2003)
• G. Preissler, Isothermic surfaces and Hopf cylinders, Beiträge Algebra Geom. 44, 1--8 (2003)
• G. Preissler, On a generalization of Willmore surfaces for hypersurfaces, Res. Math. 35, 314--324 (1999)
• F. Dillen and W. Kühnel, Ruled Weingarten surfaces in Minkowski 3-space,
  manuscripta math. 98, 307-320 (1999)
• M. van Gemmeren, Tightness of manifolds with H-spherical ends, Compositio Math. 112, 17-32 (1998)
• P. Breuer and W. Kühnel, The tightness of tubes, Forum Math. 9, 707-720 (1997)
• M. Becker and W. Kühnel, Hypersurfaces with constant inner curvature of the second fundamental form,
  and the non-rigidity of the sphere, Math. Z. 223, 693-708 (1996)
• M. van Gemmeren, Total absolute curvature and tightness of noncompact manifolds,
  Transactions Amer. Math. Soc. 348, 2413-2426 (1996)
• W. Kühnel, Ruled W-surfaces, Arch. Math. 62, 475-480 (1994)
• W. Kühnel, Tightness, torsion, and tubes, Ann. Global Analysis and Geometry 10, 227-236 (1992)
• W. Kühnel and U. Pinkall, On total mean curvatures, Quart. J. Math. Oxford (2) 37, 437-447 (1986)
• W. Kühnel and U. Pinkall, Tight smoothing of some polyhedral surfaces, in:
  Global Differential Geometry and Global Analysis, Proceedings Berlin 1984 (D. Ferus et al., eds.), 227-239,
  Lecture Notes in Mathematics 1156, Springer 1985
• W. Kühnel and T.F. Banchoff, The 9-vertex complex projective plane,
  The Math. Intelligencer Vol. 5 issue 3, 11-22 (1983)
• U. Brehm and W. Kühnel, Smooth approximation of polyhedral surfaces regarding curvatures,
  Geometriae Dedicata 12, 435-461 (1982)
• W. Kühnel, Zur inneren Krümmung der zweiten Grundform, Monatsh. Math. 91, 241-251 (1981)
• W. Kühnel, (n-2)-tightness and curvature of submanifolds with boundary,
  Intern. J. Math. Math. Sci. 1, 421-431 (1978)

On (semi-)Riemannian Geometry and Math. Physics:

• W. Kühnel and H.-B. Rademacher, Conformally Einstein product spaces,
  Diff. Geom. Appl. 49, 65--96 (2016)
• F. Leitner and A. Rod Gover, A class of compact Poincare-Einstein manifolds: properties and construction}, Comm. Contemp. Math. 12, 629--659 (2010)
• W. Kühnel and H.-B. Rademacher, Einstein spaces with a conformal group,
  Res. Math. 56, 421-444 (2009)
• A. Rod Gover and F. Leitner, A sub-product construction of Poincare-Einstein metrics,
  Intern. J. Math. 20, 1264-1287 (2009)
• F. Leitner, Conformal holonomy of bi-invariant metrics, Conform. Geom. Dyn. 12, 18-31 (2008)
• W. Kühnel and H.-B. Rademacher, Liouville's theorem in conformal geometry, J. Math. Pures et Appl. (9) 88, 251-260 (2007),
  ESI Preprint 1862 (2006) (Erwin-Schrödinger-Institut Wien), see ESI Preprints
• F. Leitner, Twistor spinors with zero on Lorentzian 5-space, Comm. Math. Phys. 275, 587-605 (2007)
• F. Leitner, On transversally symmetric pseudo-Einstein and Fefferman-Einstein spaces,
  Math. Z. 256, 443-459 (2007)
• F. Leitner, About Complex structures in conformal tractor calculus,
  ESI Preprint 1730 (2005) (Erwin-Schrödinger-Institut Wien), see ESI Preprints
• M. Steller, Conformal vector fields on spacetimes, Annals Glob. Analysis Geom. 29, 293-317 (2006)
• W. Kühnel and H.-B. Rademacher, Conformal geometry of gravitational plane waves,
  Geometriae Dedicata 109, 175-188 (2004)
• W. Kühnel and H.-B. Rademacher, Conformal Ricci collineations of space-times,
  Gen. Relativity and Gravitation 33, 1905-1914 (2001)
• W. Kühnel and H.-B. Rademacher, Asymptotically Euclidean ends of Ricci flat manifolds, and conformal inversions,
  Math. Nachrichten 219, 125-134 (2000)
• W. Kühnel and H.-B. Rademacher, Asymptotically Euclidean manifolds and twistor spinors,
  Commun. Math. Phys. 196, 67-76 (1998), Corr. ibid. 207, 735 (1999)
• W. Kühnel and H.-B. Rademacher, Twistor spinors on conformally flat manifolds,
  Illinois J. Math. 41, 495-503 (1997)
• W. Kühnel and H.-B. Rademacher, Oscillator and pendulum equation on pseudo-Riemannian spaces,
  Tôhoku Math. J. 48, 601-612 (1996)
• W. Kühnel and H.-B. Rademacher, Twistor spinors and gravitational instantons,
  Lett. Math. Phys. 38, 411-419 (1996)
• W. Kühnel and H.-B. Rademacher, Essential conformal fields in pseudo-Riemannian geometry,
  J. Math. Pures et Appl. (9) 74, 453-481 (1995), Part II in: J. Math. Sci. Univ. Tokyo 4, 649-662 (1997)
• W. Kühnel and H.-B. Rademacher, Conformal diffeomorphisms preserving the Ricci tensor,
  Proc. Amer. Math. Soc. 123, 2841-2848 (1995)
• W. Kühnel and H.-B. Rademacher, Twistor spinors with zeros, Intern. J. Math. 5, 877-895 (1994)

On Discrete and Combinatorial Geometry and Topology:

• R. Grunert, W. Kühnel and G. Rote, PL Morse theory in low dimensions, Advances in Geometry (to appear), Preprint
• U. Brehm and W. Kühnel, Lattice triangulations of E³ and of the 3-torus, Israel J. Math. 189, 97--133 (2012)
• J. Spreer, Partitioning the triangles of the cross polytope into surfaces, Beitr. Algebra und Geom. 53, 473--486 (2012)
• J. Spreer and W. Kühnel, Combinatorial properties of the K3 surface: simplicial blowups and slicings,
  Experimental Math. 20, 201-216 (2011)
• J. Spreer, Normal surfaces as combinatorial slicings, Discrete Math. 311, 1295--1309 (2011)
• F. Effenberger, Stacked polytopes and tight triangulations of manifolds, J. Comb. Th. (A) 118, 1843-1862 (2011)
• F. Effenberger and W. Kühnel, Hamiltonian submanifolds of regular polytopes ,
  Discrete Comput. Geom. 43, 242--262 (2010)
• U. Brehm and W. Kühnel, Equivelar maps on the torus, European J. Combinatorics 29, 1843-1861 (2008)
• T.F. Banchoff and W. Kühnel, Tight polyhedral models of isoparametric families, and PL--taut submanifolds ,
  Advances Geom. 7, 613-629 (2007)
• J. Itoh and W. Kühnel, Tightness of Graphs: Realizations with the Two-Piece Property,
  Rev. Roum. Math. Pures Appl. 51, 1-19 (2006)
• W. Kühnel, Tight embeddings of simply connected 4-manifolds, Documenta Math. 9, 401-412 (2004) (electronic)
• M. Casella and W. Kühnel, A triangulated K3 surface with the minimum number of vertices,
  Topology 40, 753-772 (2001)
• G. Lassmann and E. Sparla, A classification of centrally-symmetric and cyclic 12-vertex triangulations of S²x S²,
  Discrete Math. 223, 175-187 (2000)
• W. Kühnel and F. H. Lutz, A census of tight triangulations, Periodica Math. Hungarica 39, 161-183 (1999)
• E. Sparla, A new lower bound theorem for combinatorial 2k-manifolds,
  Graphs and Combinatorics 15, 109-125 (1999)
• E. Sparla, An upper and a lower bound theorem for combinatorial 4-manifolds,
  Discrete Comp. Geom. 19, 575-593 (1998)
• W. Kühnel, Topological aspects of twofold triple systems, Expositiones Math. 16, 289-331 (1998)
• W. Kühnel and G. Lassmann, Permuted difference cycles and triangulated sphere bundles,
  Discrete Math. 162, 215-227 (1996)
• U. Brehm, W. Kühnel and E. Schulte, Manifold structures on abstract regular polytopes,
  Aequationes math. 49, 12-35 (1995)
• W. Kühnel, Manifolds in the skeletons of convex polytopes, tightness, and generalized Heawood inequalities,
  Polytopes: Abstract, Convex and Computational, (T. Bisztriczky et al., eds.), pp. 241-247,
  Proc. NATO Advanced Study Institute (ASI) Ser. C Vol. 440, Kluwer 1994
• T.F. Banchoff and W. Kühnel, Equilibrium triangulations of the complex projective plane,
  Geometriae Dedicata 44, 313-333 (1992)
• U. Brehm and W. Kühnel, 15-vertex triangulations of an 8-manifold, Math. Annalen 294, 167-193 (1992)
• W. Kühnel and G. Lassmann, Combinatorial d-tori with a large symmetry group,
  Discrete Comput. Geom. 3, 169-176 (1988)
• U. Brehm and W. Kühnel, Combinatorial manifolds with few vertices, Topology 26, 465-473 (1987)
• W. Kühnel, Minimal triangulations of Kummer varieties, Abh. Math. Sem. Univ. Hamburg 57, 7-20 (1987)
• W. Kühnel and G. Lassmann, Neighborly combinatorial 3-manifolds with dihedral automorphism group,,
  Israel J. Math. 52, 147-166 (1985)
• W. Kühnel and G. Lassmann, The rhombidodecahedral tessellation of 3-space and a particular
  15-vertex triangulation of the 3-dimensional torus, manuscripta math. 49, 61-77 (1984)
• W. Kühnel and G. Lassmann, The unique 3-neighborly 4-manifold with few vertices,
  J. Comb. Th. (A) 35, 173-184 (1983)
• W. Kühnel, Tight and 0-tight polyhedral embeddings of surfaces, Inventiones math. 58, 161-177 (1980)

Related papers by other people on particular triangulations and conjectures:

• B. Morin and M. Yoshida, The Kühnel triangulation of the complex projective plane from the view-point
  of complex crystallography, Part I, Memoirs Fac. Sci. Kyushu Univ., Ser. A 45, 55-142 (1991)
• P. Arnoux and A. Marin, The Kühnel triangulation of the complex projective plane from the view-point
  of complex crystallography, Part II, Memoirs Fac. Sci. Kyushu Univ., Ser. A 45, 167 - 244 (1991)
• B. Bagchi and B. Datta, On Kühnel's 9-vertex complex projective plane, Geom. Dedicata 50, 1 - 13 (1994)
• B. Bagchi and B. Datta, Minimal triangulations of sphere bundles over the circle, J. Comb. Th. A 115, 737 - 752 (2008)
• I. Novik, Upper bound theorems for homology manifolds, Israel J. Math. 108, 45-82 (1998)
• I. Novik, On face numbers of manifolds with symmetry, Advances in Math., 192, 183-208 (2005).
• J. Chestnut, J. Sapir and Ed Swartz, Enumerative properties of triangulations of spherical bundles over S¹,
  European J. Combin. 29, 662-671 (2008),
• Ed Swartz, Face enumeration: From spheres to manifolds, J. European Math. Soc. 11, 449-485 (2009),
  see also http://www.math.cornell.edu/~ebs/papers.html

• Sommersemester 2015:
  Riemannsche Geometrie 1 (Vorlesung mit Übungen), Vorlesung Diskrete Geometrie (Vorlesung mit Übungen), Oberseminar Geometrie
• Wintersemester 2014/2015
  Differentialgeometrie (Vorlesung mit Übungen), Seminar zur Geometrie, Oberseminar Geometrie und Topologie
• Sommersememester 2014:
  Geometrie (Vorlesung mit Übungen), Oberseminar Geometrie
• Wintersemester 2013/2014:
  Geometrie (Vorlesung mit Übungen), Oberseminar Geometrie
• Sommersemester 2013:
  Algebraische Topologie 1 (Vorlesung mit Übungen), Diskrete Geometrie (Vorlesung mit Übungen), Oberseminar Geometrie
• Wintersemester 2012/2013:
  Riemannsche Geometrie 2 (Vorlesung mit Übungen), Seminar zur Riemannschen Geometrie, Oberseminar Geometrie
• Sommersemester 2012: Riemannsche Geometrie 1 (Vorlesung mit Übungen), Oberseminar Geometrie
• Wintersemester 2011/2012:
  Lie-Gruppen (Vorlesung mit Übungen), Oberseminar Geometrie
• Wintersemester 2010/2011:
  HM 3 für aer, mawi, geod (Vorlesung mit Übungen), Oberseminar Geometrie
• Wintersemester 2009/2010:
  Topologie (Vorlesung mit Übungen), Oberseminar Geometrie/td>
• Sommersemester 2009:
  Lineare Algebra und Analytische Geometrie 2 (Vorlesung mit Übungen), Oberseminar Geometrie
• Wintersemester 2008/2009:
  Lineare Algebra und Analytische Geometrie I (Vorlesung mit Übungen), Oberseminar Geometrie
• Sommersememester 2008
  Topologie II (Vorlesung mit Übungen), Differentialgeometrie für Geodäten (Vorlesung mit Übungen), Hauptseminar Topologie (gemeinsam mit Prof. Dr. E. Teufel) , Oberseminar Geometrie
• Wintersemester 2007/2008:
  Topologie (Vorlesung mit Übungen), Proseminar: Elementare Geometrie für das Lehramt, Oberseminar Geometrie
• Sommersememester 2007:
  Mathematics for Infotech, Konvexe Polytope (Vorlesung, 2-stündig), Oberseminar Geometrie
• Wintersemester 2006/2007:
  HM III für aer, verf, autip, wewi (Vorlesung mit Übungen), Oberseminar Geometrie
• Wintersemester 2005/2006:
  Differentialgeometrie (Vorlesung mit Übungen), Algebraische Topologie (Simpliziale Homologie) (Vorlesung), Oberseminar Geometrie
• Sommersememester 2005:
  Topologie (Vorlesung mit Übungen), Differentialgeometrie für Geodäten (Vorlesung mit Übungen), Oberseminar Geometrie
• Wintersemester 2004/2005:
  Lie-Gruppen (Vorlesung mit Übungen), Oberseminar Geometrie
• Sommersememester 2004:
  Riemannsche Geometrie (Vorlesung mit Übungen), Oberseminar Geometrie
• Wintersemester 2003/2004:
  Riemannsche Geometrie (Vorlesung mit Übungen), Seminar zur Geometrie, Oberseminar Geometrie

• Jonathan Spreer (2011)
  (gefördert durch eine Doktorandenstelle der DFG)
  über das Thema: Blowups, slicings and permutation groups in combinatorial topology
• Felix Effenberger (2010)
  (gefördert durch eine Doktorandenstelle der DFG)
  über das Thema: Hamiltonian submanifolds of regular polytopes
• Marc-Oliver Otto (2003)
  über das Thema: Straffe zweidimensionale Untermannigfaltigkeiten euklidischer Raumformen
• Michael Steller (2001)
  (Stipendiat der Landesgraduiertenförderung BW)
  über das Thema: Klassifikation konformer Vektorfelder mit Nullstellen auf Raumzeiten
• Gunnar Ketelhut (2001)
  über das Thema: Verallgemeinerte warped-product-Räume
• Markus Becker (1998)
  über das Thema: Konforme Gradientenvektorfelder auf Lorentz-Mannigfaltigkeiten
• Eric Sparla (1997)
  (Stipendiat der Landesgraduiertenförderung BW)
  über das Thema: Geometrische und kombinatorische Eigenschaften triangulierter Mannigfaltigkeiten
• Martin van Gemmeren (1995)
  über das Thema: Totale Krümmung und totale Absolutkrümmung von Untermannigfaltigkeiten des Rm
• Peter Breuer (1994)
  über das Thema: Röhren um straffe Immersionen