# Applying The Poincare-Hopf Theorem

In the mathematical community, it is a well-known fact, that you cannot comb a hedgehog without leaving some bald spots. Remarkably, this rather trivial insight leads to the fundamental theorem of Poincare-Hopf. This theorem states that there are several types of bald spots, called defects, endowed with a topological charge which always must sum up to the characteristic of the combed surface.

This characteristic again categorizes surfaces, neglecting their precise shape, solemn along the number of holes and their dimensionality. For example, a coffee mug has one hole (the handle), and is, therefore, topological identical to a donut, with both of them having a characteristic of 0. Another very large group of shapes are the sphere likes, including any shape in our 3D world that does not contain a hole, here the characteristic is 2.

Physical experiments observed diverse physical systems, e.g. a crystals growing on a curved substrate, which are crucially influenced by the characteristic of their domain. Physicists have coined the term geometric frustration, describing that a physical favorable state, eg regular crystal structure, could not propagate throughout the whole system leading to localized unfavorable states, eg a flaw in the otherwise regular crystal structure. To these physical defects, the concepts of Poincare Hopf theorem can also be applied.

The research on interactions of physics and the geometric properties of the domain has received fresh attention from the thriving field of biological applied physics. For example, in many biological structures, metabolic processes are controlled by the encapsulating membrane. In a mammal cell storage and transport compartments, called vesicles, are covered by an emulsion consisting of rod-like particles. These particles tend to align parallel, but due to the geometric constraints of their highly irregular but yet sphere-like shape, they cannot express such order throughout the complete surface and defects arise. Investigating the process of creation and breakdown of such vesicles indicate that defect locations on the vesicle surface are pivotal points where the breakdown of the vesicle is triggered.

In our research, we focus on a generic system consisting of rod-like particles on surfaces. Their ordering is described by a free energy defining the desired ground state where all rods are aligned parallel to each other and parallel to the surface. Any deviation from this ground state is penalized, such that defects inevitably cost energy. But due to the Poincare Hopf theorem defects could not be avoided, such that the ground state cannot be established throughout the complete surface. One aspect of our work is to develop methods for numerical simulation of such particle systems and their dynamics towards the energetic minimum. With these methods, we systematically investigate the emerging defect configurations and their dependence on the system’s geometry, more precisely the impact of local curvature. It has been established in theory and by experiments that defects, depending on their topological charge, are attracted by regions with matching curvature. Moreover, defects of opposing charge attract each other and annihilate on contact, reducing the ordering energy.

Our theoretical arguments and numeric experiments provide insights how curvature induces local forces on the defects. For sufficient strong curvature, these geometric forces outweigh the attractive forces and suppress the pairwise defect annihilation. Increasing the curvature further, we observe the creation of defect pairs. Therefore we were able to demonstrate how local modulation of curvature defines the realization of the Poincare Hopf theorem, enabling non-minimal defect configurations.

Besides these aspects, we provide theoretical arguments bridging the gap between the description of particle systems in volume(3D) and surface(2D). Considering surfaces as tubular volume with vanishing thickness shed a light how curvature does also induce a globally preferred alignment of particles along lines of minimal curvature. Among others, these results are presented in the following papers:

- M. Nestler, I. Nitschke, S. Praetorius, A. Voigt – Orientational order on surfaces – the coupling of topology, geometry, and dynamics
- I. Nitschke, M. Nestler, S. Praetorius, A. Voigt – Nematic liquid crystals on curved surfaces – a thin film limit

These findings are described in the journal article entitled Orientational Order on Surfaces: The Coupling of Topology, Geometry, and Dynamics, published in the journal Journal of Nonlinear Science. This work was led by Michael Nestler from the Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, Dresden, Germany.