Prof. Wolfgang Kühnel


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

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

Weingarten surfaces

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

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:

Conformal vector fields on space-times

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.