Differential Geometric Methods Research Articles

On the lattice spectrum contribution to the free energy of defect solids

The energy functional of lattice vibrations of a defect solid is derived. Using the ζ-function method the contribution of the lattice spectrum on defect states of solids to the free energy is then calculated perturbatively by means of differential geometric methods.

Stability and isolation phenomena for Yang-Mills fields

In this article a series of results concerning Yang-Mills fields over the euclidean sphere and other locally homogeneous spaces are proved using differential geometric methods. One of our main results is to prove that any weakly stable Yang-Mills field overS4 with groupG=SU2, SU3 orU2 is either self-dual or anti-self-dual. The analogous statement for SO4-bundles is also proved. The other main series of results concerns gap-phenomena for Yang-Mills fields. As a consequence of the non-linearity of the Yang-Mills equations, we can give explicitC0-neighbourhoods of the minimal Yang-Mills fields which contain no other Yang-Mills fields. In this part of the study the nature of the groupG does not matter, neither is the dimension of the base manifold constrained to be four.

Nonlinear Systems Analysis

Introduction. Non-linear Differential Equations. Second-Order Systems. Approximate Analysis Methods. Lyapunov Stability. Input-Output Stability. Differential Geometric Methods. Appendices: Prevalence of Differential Equations with Unique Solutions, Proof of the Kalman-Yacubovitch Lemma and Proof of the Frobenius Theorem.

Differential geometrical methods in various domains of theoretical physics

Differential geometrical methods in various domains of theoretical physics

Remarks about the thin sandwich conjecture

The thin sandwich conjecture in the initial value problem of general relativity is investigated by means of differential geometrical methods, in connection to the definition of arc length in a Wheeler superspace. The conjecture is proved to be false by good physical reasons. Its failure on a flat space is related to the “functional dimension” of the superspace and to the behaviour in the linearized theory of quantum gravitation.

Ordnungseigenschaften holomorpher Funktionen

A linear geometric order theory for holomorphic mappings F: ℂn→ℂm is given if a family of linear subspaces L of ℂn×ℂm is specified; the theory then is concerned with the order number of connected parts of the intersections L∩ (F) of the L's with the graph (F) of F. For m=1 (i.e. case of one function) it is proved that the local linear order number of a holomorphic function is finite in every point. If the L's are real-linear subspaces then real differential geometric methods lead to the proof, if the L's are complex-linear then ideal-theoretical means do.