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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
, Tôhoku Math. J. 57,
171-200 (2005)
(pdf | ps)
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
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 -space,
manuscripta math. 98, 307-320 (1999)
- W. Kühnel and M. Steller, On closed Weingarten surfaces,
Monatsh. Math. 146, 113-126 (2005)
(pdf | ps)
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) (to appear)
- W. Kühnel and H.-B. Rademacher,
Conformal transformations of pseudo-Riemannian manifolds, Preprint
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)
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