Analytic Continuation of Diagonals of Laurent Series for Rational Functions

We describe branch points of complete q-diagonals of Laurent series for rational functions in several complex variables in terms of the logarithmic Gauss mapping. The sufficient condition of non-algebraicity of such a diagonal is proven

Gauss Map Based Curved Origami Discretization

This paper addresses the problem of discretizing the curved developable surfaces that are satisfying the equivalent surface curvature change discretizations. Solving basic folding units occurs in such tasks as simulating the behavior of Gauss mapping. The Gauss spherical curves of different developable surfaces are setup under the Gauss map. Gauss map is utilized to investigate the normal curvature change of the curved surface. In this way, spatial curved surfaces are mapped to spherical curves. Each point on the spherical curve represents a normal direction of a ruling line on the curved surface. This leads to the curvature discretization of curved surface being transferred to the normal direction discretization of spherical curves. These developable curved surfaces are then discretized into planar patches to acquire the geometric properties of curved folding such as fold angle, folding direction, folding shape, foldability, and geometric constraints of adjacent ruling lines. It acts as a connection of curved and straight folding knowledge. The approach is illustrated in the context of the Gauss map strategy and the utility of the technique is demonstrated with the proposed principles of Gauss spherical curves. It is applicable to any generic developable surfaces.

VARIETIES WITH DEGENERATE GAUSS MAPPINGS IN COMPLEX HYPERBOLIC SPACE FORMS

In analogy with the Gauss mapping for a subvariety in the complex projective space, the Gauss mapping for a subvariety in a complex hyperbolic space form can be defined as a map from the smooth locus of the subvariety to the quotient of a suitable domain in the Grassmannian. For complex hyperbolic space forms of finite volume, it is proved that the Gauss mapping is degenerate if and only if the subvariety is totally geodesic.

Geometrically Induced Gauge Structure on Manifolds Embedded in a Higher-Dimensional Space

We explain in a context different from that of Maraner the formalism for describing the motion of a particle, under the influence of a confining potential, in a neighborhood of an n-dimensional curved manifold Mn embedded in a p-dimensional Euclidean space Rp with p ≥ n + 2. The effective Hamiltonian on Mn has a (generally non-Abelian) gauge structure determined by the geometry of Mn. Such a gauge term is defined in terms of the vectors normal to Mn, and its connection is called the N connection. This connection is nothing but the connection induced from the normal connection of the submanifold Mn of Rp. In order to see the global effect of this type of connections, the case of M1 embedded in R3 is examined, where the relation of an integral of the gauge potential of the N connection (i.e. the torsion) along a path in M1 to the Berry phase is given through Gauss mapping of the vector tangent to M1. Through the same mapping in the case of M1 embedded in Rp, where the normal and the tangent quantities are exchanged, the relation of the N connection to the induced gauge potential (the canonical connection of the second kind) on the (p - 1)-dimensional sphere Sp - 1 (p ≥ 3) found by Ohnuki and Kitakado is concretely established; the former is the pullback of the latter by the Gauss mapping. Further, this latter which has the monopole-like structure is also proved to be gauge-equivalent to the spin connection of Sp - 1. Thus the N connection is also shown to coincide with the pullback of the spin connection of Sp - 1. Finally, by extending formally the fundamental equations for Mn to the infinite-dimensional case, the present formalism is applied to the field theory that admits a soliton solution. The resultant expression is in some respects different from that of Gervais and Jevicki.