The geometric Cauchy problem for developable submanifolds

Given a smooth distribution $\mathscr{D}$ of $m$-dimensional planes along a smooth regular curve $\gamma$ in $\mathbb{R}^{m+n}$, we consider the following problem: To find an $m$-dimensional developable submanifold of $\mathbb{R}^{m+n}$, that is, a ruled submanifold with constant tangent space along the rulings, such that its tangent bundle along $\gamma$ coincides with $\mathscr{D}$. In particular, we give sufficient conditions for the local well-posedness of the problem, together with a parametric description of the solution.Comment: 15 page

Cartan Ribbonization of Surfaces and a Topological Inspection

We develop the concept of Cartan ribbons and a method by which they can be used to ribbonize any given surface in space by intrinsically flat ribbons. The geodesic curvature along the center curve on the surface agrees with the geodesic curvature of the corresponding Cartan development curve, and this makes a rolling strategy successful. Essentially, it follows from the orientational alignment of the two co-moving Darboux frames during the rolling. Using closed center curves we obtain closed approximating Cartan ribbons that contribute zero to the total curvature integral of the ribbonization. This paves the way for a particular simple topological inspection -- it is reduced to the question of how the ribbons organize their edges relative to each other. The Gauss-Bonnet theorem leads to this topological inspection of the vertices. Finally, we display two examples of ribbonizations of surfaces, namely of a torus using two ribbons, and of an ellipsoid using its closed curvature lines as center curves for the ribbons. The topological inspection of the torus ribbonization is particularly simple as it has no vertex points, giving directly the Euler characteristic $0$. The ellipsoid has $4$ vertices -- corresponding to the $4$ umbilical points -- each of degree one and each therefore contributing one-half to the Euler characteristic

Nonrigidity of flat ribbons

We study ribbons of vanishing Gaussian curvature, i.e., flat ribbons, constructed along a curve in $\mathbb{R}^{3}$. In particular, we first investigate to which extent the ruled structure determines a flat ribbon: in other words, we ask whether for a given curve $\gamma$ and ruling angle (angle between the ruling line and the curve's tangent) there exists a well-defined flat ribbon. It turns out that the answer is positive only up to an initial condition, expressed by a choice of normal vector at a point. We then study the set of infinitely narrow flat ribbons along a fixed curve $\gamma$ in terms of energy. By extending a well-known formula for the bending energy of the rectifying developable, introduced in the literature by Sadowsky in 1930, we obtain an upper bound for the ratio between the bending energies of two solutions of the initial value problem. We finally draw further conclusions under some additional assumptions on the ruling angle and the curve $\gamma$.Comment: 16 pages, 3 figure

Total torsion of three-dimensional lines of curvature

A curve $\gamma$ in a Riemannian manifold $M$ is three-dimensional if its torsion (signed second curvature function) is well-defined and all higher-order curvatures vanish identically. In particular, when $\gamma$ lies on an oriented hypersurface $S$ of $M$, we say that $\gamma$ is well positioned if the curve's principal normal, its torsion vector, and the surface normal are everywhere coplanar. Suppose that $\gamma$ is three-dimensional and closed. We show that if $\gamma$ is a well-positioned line of curvature of $S$, then its total torsion is an integer multiple of $2\pi$; and that, conversely, if the total torsion of $\gamma$ is an integer multiple of $2\pi$, then there exists an oriented hypersurface of $M$ in which $\gamma$ is a well-positioned line of curvature. Moreover, under the same assumptions, we prove that the total torsion of $\gamma$ vanishes when $S$ is convex. This extends the classical total torsion theorem for spherical curves.Comment: 8 pages, no figures. Accepted versio

Dynamic instability in a phenomenological model of correlated assets

We show that financial correlations exhibit a non-trivial dynamic behavior. We introduce a simple phenomenological model of a multi-asset financial market, which takes into account the impact of portfolio investment on price dynamics. This captures the fact that correlations determine the optimal portfolio but are affected by investment based on it. We show that such a feedback on correlations gives rise to an instability when the volume of investment exceeds a critical value. Close to the critical point the model exhibits dynamical correlations very similar to those observed in real markets. Maximum likelihood estimates of the model’s parameter for empirical data indeed confirm this conclusion, thus suggesting that real markets operate close to a dynamically unstable point

Risk bubbles and market instability

We discuss a simple model of correlated assets capturing the feedback effects induced by portfolio investment in the covariance dynamics. This model predicts an instability when the volume of investment exceeds a critical value. Close to the critical point the model exhibits dynamical correlations very similar to those observed in real markets. Maximum likelihood estimates of the model’s parameter for empirical data indeed confirms this conclusion. We show that this picture is confirmed by the empirical analysis for different choices of the time horizon

Curvature adapted submanifolds of bi-invariant Lie groups

We study submanifolds of arbitrary codimension in a Lie group $\mathsf{G}$ equipped with a bi-invariant metric. In particular, we show that, if the normal bundle of $M \subset \mathsf{G}$ is abelian, then the normal Jacobi operator of $M$ equals the square of its invariant shape operator. This allows us to obtain geometric conditions which are necessary and sufficient for the submanifold $M$ to be curvature adapted to $\mathsf{G}$

Planar pseudo-geodesics and totally umbilic submanifolds

We study totally umbilic isometric immersions between Riemannian manifolds. First, we provide a novel characterization of the totally umbilic isometric immersions with parallel normalized mean curvature vector, i.e., those having nonzero mean curvature vector and such that the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. Such characterization is based on a family of curves, called planar pseudo-geodesics, representing a natural extrinsic generalization of both geodesics and Riemannian circles: being planar, their Cartan development in the tangent space is planar in the ordinary sense; being pseudo-geodesics, their geodesic and normal curvatures satisfy a linear relation. We study these curves in detail and, in particular, establish their local existence and uniqueness. Moreover, in the case of codimension-one immersions, we prove the following statement: an isometric immersion $\iota \colon M \hookrightarrow Q$ is totally umbilic if and only if the extrinsic shape of every geodesic of $M$ is planar. This extends a well-known result about surfaces in $\mathbb{R}^{3}$.Comment: 15 pages, no figures. Significant changes in sections 1, 3, 4, and