Generalized DPW method and an application to isometric immersions of space forms

Let $G$ be a complex Lie group and $\Lambda G$ denote the group of maps from the unit circle ${\mathbb S}^1$ into $G$, of a suitable class. A differentiable map $F$ from a manifold $M$ into $\Lambda G$, is said to be of \emph{connection order $(_a^b)$} if the Fourier expansion in the loop parameter $\lambda$ of the ${\mathbb S}^1$-family of Maurer-Cartan forms for $F$, namely F_\lambda^{-1} \dd F_\lambda, is of the form $\sum_{i=a}^b \alpha_i \lambda^i$. Most integrable systems in geometry are associated to such a map. Roughly speaking, the DPW method used a Birkhoff type splitting to reduce a harmonic map into a symmetric space, which can be represented by a certain order $(_{-1}^1)$ map, into a pair of simpler maps of order $(_{-1}^{-1})$ and $(_1^1)$ respectively. Conversely, one could construct such a harmonic map from any pair of $(_{-1}^{-1})$ and $(_1^1)$ maps. This allowed a Weierstrass type description of harmonic maps into symmetric spaces. We extend this method to show that, for a large class of loop groups, a connection order $(_a^b)$ map, for $a<0<b$, splits uniquely into a pair of $(_a^{-1})$ and $(_1^b)$ maps. As an application, we show that constant non-zero curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds, reducing the problem (at least locally) to the flat case. To extend the DPW method sufficiently to handle this problem requires a more general Iwasawa type splitting of the loop group, which we prove always holds at least locally.Comment: Some typographical correction

Time-resolved Fourier transform infrared emission spectroscopy of CO ∆v = 1 and ∆v = 2 extended bands in the ground X1Σ+ state produced by formamide glow discharge

This paper presents an extension to our knowledge of ∆v = 1 and ∆v = 2 bands of carbon monoxide in the ground state, measured by Fourier transform infrared spectroscopy of glow discharge of formamide-nitrogen mixture. Lines in declared bands are measured up to v = 30 for ∆v = 1 and up to v = 24 for ∆v = 2 band, by use of both InSb and MCT detectors, which have not been measured in the laboratory before. Dunham parameters obtained by fitting our lines are presented as well as comparison to other authors. The paper also demonstrates the interesting impossibility of sufficient population of ∆v = 2 band of CO when only pure CO is used in the glow discharge, instead of formamide-based mixture. Additionally, we present a non-LTE model to describe the intensity pattern of the ∆v = 1 and the ∆v = 2 bands of 12C16O experimental spectra by simulating the corresponding non-LTE vibrational populations of CO

Limitations on the smooth confinement of an unstretchable manifold

We prove that an m-dimensional unit ball D^m in the Euclidean space {\mathbb R}^m cannot be isometrically embedded into a higher-dimensional Euclidean ball B_r^d \subset {\mathbb R}^d of radius r < 1/2 unless one of two conditions is met -- (1)The embedding manifold has dimension d >= 2m. (2) The embedding is not smooth. The proof uses differential geometry to show that if d<2m and the embedding is smooth and isometric, we can construct a line from the center of D^m to the boundary that is geodesic in both D^m and in the embedding manifold {\mathbb R}^d. Since such a line has length 1, the diameter of the embedding ball must exceed 1.Comment: 20 Pages, 3 Figure

Polar foliations and isoparametric maps

A singular Riemannian foliation $F$ on a complete Riemannian manifold $M$ is called a polar foliation if, for each regular point $p$, there is an immersed submanifold $\Sigma$, called section, that passes through $p$ and that meets all the leaves and always perpendicularly. A typical example of a polar foliation is the partition of $M$ into the orbits of a polar action, i.e., an isometric action with sections. In this work we prove that the leaves of $F$ coincide with the level sets of a smooth map $H: M\to \Sigma$ if $M$ is simply connected. In particular, we have that the orbits of a polar action on a simply connected space are level sets of an isoparametric map. This result extends previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter and West, and Terng.Comment: 9 pages; The final publication is available at springerlink.com http://www.springerlink.com/content/c72g4q5350g513n1

Differential systems associated with tableaux over Lie algebras

We give an account of the construction of exterior differential systems based on the notion of tableaux over Lie algebras as developed in [Comm. Anal. Geom 14 (2006), 475-496; math.DG/0412169]. The definition of a tableau over a Lie algebra is revisited and extended in the light of the formalism of the Spencer cohomology; the question of involutiveness for the associated systems and their prolongations is addressed; examples are discussed.Comment: 16 pages; to appear in: "Symmetries and Overdetermined Systems of Partial Differential Equations" (M. Eastwood and W. Miller, Jr., eds.), IMA Volumes in Mathematics and Its Applications, Springer-Verlag, New Yor

A factorization of a super-conformal map

A super-conformal map and a minimal surface are factored into a product of two maps by modeling the Euclidean four-space and the complex Euclidean plane on the set of all quaternions. One of these two maps is a holomorphic map or a meromorphic map. These conformal maps adopt properties of a holomorphic function or a meromorphic function. Analogs of the Liouville theorem, the Schwarz lemma, the Schwarz-Pick theorem, the Weierstrass factorization theorem, the Abel-Jacobi theorem, and a relation between zeros of a minimal surface and branch points of a super-conformal map are obtained.Comment: 21 page

Singular riemannian foliations with sections, transnormal maps and basic forms

A singular riemannian foliation F on a complete riemannian manifold M is said to admit sections if each regular point of M is contained in a complete totally geodesic immersed submanifold (a section) that meets every leaf of F orthogonally and whose dimension is the codimension of the regular leaves of F. We prove that the algebra of basic forms of M relative to F is isomorphic to the algebra of those differential forms on a section that are invariant under the generalized Weyl pseudogroup of this section. This extends a result of Michor for polar actions. It follows from this result that the algebra of basic function is finitely generated if the sections are compact. We also prove that the leaves of F coincide with the level sets of a transnormal map (generalization of isoparametric map) if M is simply connected, the sections are flat and the leaves of F are compact. This result extends previous results due to Carter and West, Terng, and Heintze, Liu and Olmos.Comment: Preprint IME-USP; The final publication is available at springerlink.com http://www.springerlink.com/content/q48682633730t831

Progress in the Theory of Singular Riemannian Foliations

A singular foliation is called a singular Riemannian foliation (SRF) if every geodesic that is perpendicular to one leaf is perpendicular to every leaf it meets. A typical example is the partition of a complete Riemannian manifold into orbits of an isometric action. In this survey, we provide an introduction to the theory of SRFs, leading from the foundations to recent developments in research on this subject. Sketches of proofs are included and useful techniques are emphasized. We study the local structure of SRFs in general and under curvature conditions. We review the solution of the Palais-Terng problem on integrability of the horizontal distribution. Important special classes of SRFs, like polar and variationally complete foliations and their connections, are treated. A characterisation of SRFs whose leaf space is an orbifold is given. Moreover, desingularizations of SRFs are studied and applications, e.g., to Molino's conjecture, are presented

Identifiable Acetylene Features Predicted for Young Earth-like Exoplanets with Reducing Atmospheres Undergoing Heavy Bombardment

The chemical environments of young planets are assumed to be largely influenced by the impacts of bodies lingering on unstable trajectories after the dissolution of the protoplanetary disk. We explore the chemical consequences of impacts within the context of reducing planetary atmospheres dominated by carbon monoxide, methane, and molecular nitrogen. A terawatt high-power laser was selected in order to simulate the airglow plasma and blast wave surrounding the impactor. The chemical results of these experiments are then applied to a theoretical atmospheric model. The impact simulation results in substantial volume mixing ratios within the reactor of 5% hydrogen cyanide (HCN), 8% acetylene (C2H2), 5% cyanoacetylene (HC3N), and 1% ammonia (NH3). These yields are combined with estimated impact rates for the early Earth to predict surface boundary conditions for an atmospheric model. We show that impacts might have served as sources of energy that would have led to steady-state surface quantities of 0.4% C2H2, 400 ppm HCN, and 40 ppm NH3. We provide simulated transit spectra for an Earth-like exoplanet with this reducing atmosphere during and shortly after eras of intense impacts. We predict that acetylene is as observable as other molecular features on exoplanets with reducing atmospheres that have recently gone through their own "heavy bombardments," with prominent features at 3.05 and 10.5 μm