Browsing by Author "Bryant, RL"

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Open Access
A solution of a problem of Sophus Lie: Normal forms of two-dimensional metrics admitting two projective vector fields
(Mathematische Annalen, 2008-02-01) Bryant, RL; Manno, G; Matveev, VS
We give a complete list of normal forms for the two-dimensional metrics that admit a transitive Lie pseudogroup of geodesic-preserving transformations and we show that these normal forms are mutually non-isometric. This solves a problem posed by Sophus Lie. © 2007 Springer-Verlag.
Open Access
Bochner-kähler metrics
(Journal of the American Mathematical Society, 2001-07-01) Bryant, RL
Open Access
Calibrated Embeddings in the Special Lagrangian and Coassociative Cases
(Annals of Global Analysis and Geometry, 2000-12-01) Bryant, RL
Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G2-manifold, even as the fixed locus of an anti-G2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated sub-manifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.
Open Access
Calibrated embeddings in the special Lagrangian and coassociative cases
(Annals of Global Analysis and Geometry, 2000) Bryant, RL
Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.
Open Access
D-branes and Spinc structures
(Physics Letters, Section B: Nuclear, Elementary Particle and High-Energy Physics, 1999-12-01) Sharpe, E; Bryant, RL
It was recently pointed out by E. Witten that for a D-brane to consistently wrap a submanifold of some manifold, the normal bundle must admit a Spinc structure. We examine this constraint in the case of type II string compactifications with vanishing cosmological constant, and argue that in all such cases, the normal bundle to a supersymmetric cycle is automatically Spinc. © 1999 Published by Elsevier Science B.V. All rights reserved.
Open Access
Geodesic behavior for Finsler metrics of constant positive flag curvature on $S^2$
(2017-11-01) Bryant, RL; Foulon, P; Ivanov, SV; Matveev, VS; Ziller, W
We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metrics with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on~$S^2$ with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or have constant flag curvature
Open Access
Gradient Kahler Ricci solitons
(Asterisque, 2008-10-01) Bryant, RL
Some observations about the local and global generality of gradient Kahler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a prescribed holomorphic volume form and vector field, and the existence of Poincaré coordinates in the case that the Ricci curvature is positive and the vector field has a fixed point. © Asterisque 321.
Open Access
Non-Embedding and Non-Extension Results in Special Holonomy
(The Many Facets of Geometry: A Tribute to Nigel Hitchin, 2010-09-01) Bryant, RL
© Oxford University Press 2010. All rights reserved.In the early analyses of metrics with special holonomy in dimensions 7 and 8, particularly in regards to their existence and generality, heavy use was made of the Cartan-Kähler theorem, essentially because the analyses were reduced to the study of overdetermined PDE systems whose natures were complicated by their diffeomorphism invariance. The Cartan-Kähler theory is well suited for the study of such systems and the local properties of their solutions. However, the Cartan-Kähler theory is not particularly well suited for studies of global problems for two reasons: first, it is an approach to PDE that relies entirely on the local solvability of initial value problems and, second, the Cartan-Kähler theory is only applicable in the real-analytic category. Nevertheless, when there are no other adequate methods for analyzing the local generality of such systems, the Cartan-Kähler theory is a useful tool and it has the effect of focusing attention on the initial value problem as an interesting problem in its own right. This chapter clarifies some of the existence issues involved in applying the initial value problem to the problem of constructing metrics with special holonomy. In particular, it discusses the role of the assumption of real-analyticity and presents examples to show that one cannot generally avoid such assumptions in the initial value formulations of these problems.
Open Access
Notes on exterior differential systems
(2014-05-13) Bryant, RL
These are notes for a very rapid introduction to the basics of exterior differential systems and their connection with what is now known as Lie theory, together with some typical and not-so-typical applications to illustrate their use.
Open Access
On Finsler surfaces of constant flag curvature with a Killing field
(Journal of Geometry and Physics, 2017-06-01) Bryant, RL; Huang, L; Mo, X
© 2017 Elsevier B.V. We study two-dimensional Finsler metrics of constant flag curvature and show that such Finsler metrics that admit a Killing field can be written in a normal form that depends on two arbitrary functions of one variable. Furthermore, we find an approach to calculate these functions for spherically symmetric Finsler surfaces of constant flag curvature. In particular, we obtain the normal form of the Funk metric on the unit disk D 2 .
Open Access
On the conformal volume of 2-tori
(2015-07-06) Bryant, RL
This note provides a proof of a 1985 conjecture of Montiel and Ros about the conformal volume of tori. (This material is not really new; I'm making it available now because of requests related to recent interest in the conjecture.)
Open Access
On the convex Pfaff-Darboux Theorem of Ekeland and Nirenberg
(2015-12-22) Bryant, RL
The classical Pfaff-Darboux Theorem, which provides local `normal forms' for 1-forms on manifolds, has applications in the theory of certain economic models. However, the normal forms needed in these models come with an additional requirement of convexity, which is not provided by the classical proofs of the Pfaff-Darboux Theorem. (The appropriate notion of `convexity' is a feature of the economic model. In the simplest case, when the economic model is formulated in a domain in n-space, convexity has its usual meaning. In 2002, Ekeland and Nirenberg were able to characterize necessary and sufficient conditions for a given 1-form to admit a convex local normal form (and to show that some earlier attempts at this characterization had been unsuccessful). In this article, after providing some necessary background, I prove a strengthened and generalized convex Pfaff-Darboux Theorem, one that covers the case of a Legendrian foliation in which the notion of convexity is defined in terms of a torsion-free affine connection on the underlying manifold. (The main result in Ekeland and Nirenberg's paper concerns the case in which the affine connection is flat.)
Open Access
Projectively flat finsler 2-spheres of constant curvature
(Selecta Mathematica, New Series, 1997-12-01) Bryant, RL
After recalling the structure equations of Finsler structures on surfaces, I define a notion of "generalized Finsler structure" as a way of microlocalizing the problem of describing Finsler structures subject to curvature conditions. I then recall the basic notions of path geometry on a surface and define a notion of "generalized path geometry" analogous to that of "generalized Finsler structure." I use these ideas to study the geometry of Finsler structures on the 2-sphere that have constant Finsler-Gauss curvature K and whose geodesic path geometry is projectively flat, i.e., locally equivalent to that of straight lines in the plane. I show that, modulo diffeomorphism, there is a 2-parameter family of projectively flat Finsler structures on the sphere whose Finsler-Gauss curvature K is identically 1. © Birkhäuser Verlag, 1997.
Open Access
Real hypersurfaces in unimodular complex surfaces
(2004-07-27) Bryant, RL
A unimodular complex surface is a complex 2-manifold X endowed with a holomorphic volume form. A strictly pseudoconvex real hypersurface M in X inherits not only a CR-structure but a canonical coframing as well. In this article, this canonical coframing on M is defined, its invariants are discussed and interpreted geometrically, and its basic properties are studied. A natural evolution equation for strictly pseudoconvex real hypersurfaces in unimodular complex surfaces is defined, some of its properties are discussed, and several examples are computed. The locally homogeneous examples are determined and used to illustrate various features of the geometry of the induced structure on the hypersurface.
Open Access
Recent advances in the theory of holonomy
(Asterisque, 2000-12-01) Bryant, RL
After its introduction by Élie Cartan, the notion of holonomy has become increasingly important in Riemannian and affine geometry. Beginning with the fundamental work of Marcel Berger, the classification of possible holonomy groups of torsion free connections, either Riemannian or affine, has continued to be developed, with major breakthroughs in the last ten years. I will report on the local classification in the affine case, Joyce's fundamental work on compact manifolds with exceptional holonomies and their associated geometries, and some new work on the classification of holonomies of connections with restricted torsion, which has recently become of interest in string theory.
Open Access
Rigidity and quasi-rigidity of extremal cycles in Hermitian symmetric spaces
(2001-03-05) Bryant, RL
I use local differential geometric techniques to prove that the algebraic cycles in certain extremal homology classes in Hermitian symmetric spaces are either rigid (i.e., deformable only by ambient motions) or quasi-rigid (roughly speaking, foliated by rigid subvarieties in a nontrivial way). These rigidity results have a number of applications: First, they prove that many subvarieties in Grassmannians and other Hermitian symmetric spaces cannot be smoothed (i.e., are not homologous to a smooth subvariety). Second, they provide characterizations of holomorphic bundles over compact Kahler manifolds that are generated by their global sections but that have certain polynomials in their Chern classes vanish (for example, c_2 = 0, c_1c_2 - c_3 = 0, c_3 = 0, etc.).
Open Access
S.-S. Chern's study of almost-complex structures on the six-sphere
(2014-05-14) Bryant, RL
In 2003, S.-s. Chern began a study of almost-complex structures on the 6-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group $G_2$. While he did not solve the (currently still open) problem of determining whether there exists an integrable almost-complex structure on the 6-sphere, he did prove a significant identity that resolves the question for an interesting class of almost-complex structures on the 6-sphere.
Open Access
SO(n)-Invariant special Lagrangian submanifolds of ℂ n+1 with fixed loci
(Chinese Annals of Mathematics. Series B, 2006-01-01) Bryant, RL
Let SO(n) act in the standard way on ℂn and extend this action in the usual way to ℂn+1 = ℂ ⊕ ℂ n . It is shown that a nonsingular special Lagrangian submanifold L ⊂ ℂn+1 that is invariant under this SO(n)-action intersects the fixed ℂ ⊂ ℂ n+1 in a nonsingular real-analytic arc A (which may be empty). If n > 2, then A has no compact component. Conversely, an embedded, noncompact nonsingular real-analytic arc A ⊂ ℂ lies in an embedded nonsingular special Lagrangian submanifold that is SO(n)-invariant. The same existence result holds for compact A if n = 2. If A is connected, there exist n distinct nonsingular SO(n)-invariant special Lagrangian extensions of A such that any embedded nonsingular SO(n)-invariant special Lagrangian extension of A agrees with one of these n extensions in some open neighborhood of A. The method employed is an analysis of a singular nonlinear PDE and ultimately calls on the work of Gérard and Tahara to prove the existence of the extension. © The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2006.
Open Access
Some differential complexes within and beyond parabolic geometry
(2012-03-19) Bryant, RL; Eastwood, MG; Gover, AR; Neusser, K
For smooth manifolds equipped with various geometric structures, we construct complexes that replace the de Rham complex in providing an alternative fine resolution of the sheaf of locally constant functions. In case that the geometric structure is that of a parabolic geometry, our complexes coincide with the Bernstein-Gelfand-Gelfand complex associated with the trivial representation. However, at least in the cases we discuss, our constructions are relatively simple and avoid most of the machinery of parabolic geometry. Moreover, our method extends to certain geometries beyond the parabolic realm.
Open Access
Some examples of special Lagrangian Tori
(Advances in Theoretical and Mathematical Physics, 1999-01-01) Bryant, RL