An Exactly Solvable Case for a Thin Elastic Rod

We present a new exact solution for the twist of an asymmetric thin elastic rods. The shape of such rods is described by the static Kirchhoff equations. In the case of constant curvatire and torsion the twist of the asymmetric rod represents a soliton lattice.Comment: 6 pages, title changed, revised versio

Geometry induced potential on a 2D-section of a wormhole: catenoid

We show that a two dimensional wormhole geometry is equivalent to a catenoid, a minimal surface. We then obtain the curvature induced geometric potential and show that the ground state with zero energy corresponds to a reflectionless potential. By introducing an appropriate coordinate system we also obtain bound states for different angular momentum channels. Our findings can be realized in suitably bent bilayer graphene sheets with a neck or in a honeycomb lattice with an array of dislocations or in nanoscale waveguides in the shape of a catenoid.Comment: to appear in Phys.Rev.

Quantum anticentrifugal force for wormhole geometry

We show the existence of an anticentrifugal force in a wormhole geometry in $R^3$. This counterintuitive force was shown to exist in a flat $R^2$ space. The role the geometry plays in the appearance of this force is discussed.Comment: to appear in Physics Letters

Intrinsic twisted geometry underlying topological invariants

Topological invariants such as winding numbers and linking numbers appear in diverse physical systems described by a three-component unit vector field defined on two and three dimensional manifolds. We map the vector field to the tangent of a space curve, use a rotated Frenet-Serret frame on it, and depict the physical manifolds in terms of evolving space curves. Invoking the concept of parallel transport and the associated anholonomy (or geometric phase), we show that these topological invariants can be written as integrals of certain {\em intrinsic} geometric quantities. Our results are similar to the Gauss-Bonnet relationship which shows that the Euler characteristic (a topological invariant) is an integral of the Gaussian curvature, an intrinsic geometric quantity. For the winding number in two dimensions, these quantities are torsions of the evolving space curves, signifying their nonplanarity. In three dimensions, in addition to torsions, intrinsic twists of the space curves are necessary to obtain a nontrivial winding number and linking number. An application of our results to a 3D Heisenberg ferromagnetic model is given.Comment: 23 pages, 1 figure, 1 tabl