The local defect index up to finite ambiguity

The rational Hopf invariant of the primary obstruction to a nowhere vanishing section in a vectorbundle is computed in terms of characteristic classes. This leads to a local rigidity result on the local defect index of a map

Tension field and Index form of Energy-Type Functionals

We derive variational formulae for natural first order energy function- als and obtain criteria for the stability of isometric immersions. This generalizes known results for the classical energy, the p-energy and the exponential energ

Homotopy classes with small Jacobians

If the infimum of the conformal k-Jacobian on the homotopy class of a map between compact Riemannian manifolds vanishes then the map factors rationally through the k-skeleton of the target manifold

Stability of natural energy functionals at Riemannian subimmersions

We derive variational formulas of natural first order functionals and obtain criteria for stability in particular at Riemannian subimmersions

The local defect index up to finite ambiguity

The rational Hopf invariant of the primary obstruction to a nowhere vanishing section in a vectorbundle is computed in terms of characteristic classes. This leads to a local rigidity result on the local defect index of a map

The Global Defect Index

We show how far the local defect index determines the behaviour of an ordered medium in the vicinity of a defect

Manifolds Carrying Large Scalar Curvature

Let W = S E be a complex spinor bundle with vanishing first Chern class over a simply connected spin manifold M of dimension 5. Up to connected sums we prove that W admits a twisted Dirac operator with positive order-0-term in the Weitzenb¨ock decomposition if and only if the characteristic numbers Aˆ(TM)[M] and ch (E)Aˆ(TM)[M] vanish. This is achieved by generalizing [2] to twisted Dirac operators

Bordism of regularly defective maps

To a topological space V we assign the bordism group Ndef n (V ) of reg- ularly defective maps f : M◦→V on closed n-dimensional manifolds M. These are triples (M,, f) where is a closed submanifold ⊂ M and f a continuous map f : M r → V . We briefly review the construction of the defect complex DV given by M. Rost in [17] and show that Ndef n (V ) is isomorphic to ordinary bordism Nn(DV ). The bordism classes in Ndef n (V ) ∼= Nn(DV ) are detected by characteristic numbers twisted with cohomology classes of DV . Some of these numbers can be described without reference to the defect complex. As an example we treat the case of the circle V = S1. We compute Ndef n (S1), construct a basis and a complete set of characteristic numbers

An Integral Formula for the Maslov-Index of a Symplectic Path

The Maslov index of a not necessarily closed path M in the symplectic group Sp(2n) is expressed by an integral formula. We have an explicit formula for the integrand which is a rational 1-form on Sp(2n)

Manifolds Carrying Large Scalar Curvature

Let W = S E be a complex spinor bundle with vanishing first Chern class over a simply connected spin manifold M of dimension 5. Up to connected sums we prove that W admits a twisted Dirac operator with positive order-0-term in the Weitzenb¨ock decomposition if and only if the characteristic numbers Aˆ(TM)[M] and ch (E)Aˆ(TM)[M] vanish. This is achieved by generalizing [2] to twisted Dirac operators